Oscillatory dynamics of a metallic projectile insidea coil gun

Abstract

A coil gun is a popular electromagnetic launch system and an engaging physics experiment for undergraduate students. In a typical coil gun apparatus, a metallic projectile is launched from one end of a solenoid after traveling in a single direction through its interior. Here, we present a variation of the traditional coil gun experiment in which a spherical steel projectile undergoes oscillatory motion inside the solenoid under the influence of its magnetic field. We measure the projectile’s exit velocity as a function of the duration of the current pulse driving the solenoid and propose a theoretical model to describe its motion. This experiment is an accessible and fun physics project that explores fundamental concepts in electromagnetism, introduces richer dynamical behavior to a classic experiment, and is well suited for undergraduate teaching laboratories.

Keywords:
Electromagnetism; Experimental Physics; Physics Teaching Laboratory

1. Introduction

A coil gun is a device that uses the magnetic field generated by a solenoid to propel a metallic projectile [¹]. The coil is connected to an external electrical circuit, and a current pulse produces a magnetic field inside the solenoid. The magnetic field gradient at the ends of the finite solenoid gives rise to an electromagnetic force that attracts and accelerates the projectile. To maximize the projectile’s acceleration, the pulse duration should be sufficiently long to allow the projectile to traverse the region where the magnetic field gradient is significant. However, if the pulse persists after the projectile passes the center of the solenoid, the direction of the magnetic force reverses, decelerating the projectile. Repelled by the magnetic force near the opposite end of the solenoid, the projectile experiences a reduction in its exitvelocity.

To maximize the projectile’s exit velocity, the current energizing the solenoid must be switched off before the projectile reaches the solenoid’s center. Typically, a capacitor bank is employed to generate short, high-intensity current pulses [¹, ², ³]. To achieve higher exit velocities, multiple coil stages may be used [¹, ⁴, ⁵, ⁶, ⁷, ⁸]. Alternatively, the direction of the current can be reversed after the projectile passes the solenoid’s center [⁶]. Although the physics of coil guns is well understood, considerable effort is devoted to optimizing the coil design [⁶, ⁹, ¹⁰], capacitor discharge timing [⁸, ¹¹], and projectile’s shape [⁴] and material [¹²] in order to maximize the exit velocity.

Coil guns have multiple applications, including non-lethal defense systems [¹³], space launch technologies, [⁸, ¹⁴] vehicle propulsion [¹⁵], and forensic investigations [¹⁶]. In addition, coil guns constitute a fun and engaging physics experiment that offers valuable opportunities for the study of electromagnetism [², ³, ⁶, ¹²].

In this paper, we describe an investigation into the dynamics of a ferromagnetic projectile inside a solenoid as a function of the current pulse duration. Typically, in a coil gun, the projectile accelerates and moves in a single direction inside the solenoid, towards its exit end. In contrast, we report here the experimental observation of a richer dynamics consisting of oscillatory motion of the projectile inside the solenoid. A theoretical model for the motion of the projectile is presented. Although oscillatory motion of the projectile inside the solenoid may not be of practical interest in coil-gun applications, the optimization of the driving-current pulse duration to maximize the projectile’s launch velocity, as described here, may be. Nevertheless, such oscillatory motion of the projectile is interesting physics well suited to exploration in undergraduate teaching laboratories, such as in project-based laboratories where students design and wind their own solenoid [¹⁷].

The paper is organized as follows. In Section 2, we describe the theoretical model for the coil gun circuitry and the model for the motion of the projectile inside the solenoid. Section 3 presents the experimental setup. The theoretical and experimental results are presented and discussed in Section 4. We conclude in Section 5.

2. Theoretical Model

There are two aspects to modeling the coil gun and the dynamics of the projectile inside it. First, the electrical behavior must be described, namely the current in the circuit that drives the coil. Second, the mechanical motion of the projectile must be modeled by describing the forces acting on it.

2.1. Circuit model

Usually, a coil gun is modeled as an RLC circuit [², ³, ⁶]. To produce current pulses, we will use a pulse generator, instead of a capacitor. Therefore, our coil gun system is better viewed as a circuit in which the coil gun is represented as an inductor ($L$) and a resistor ($R$) driven by the pulse generator (Figure 1).

Figure 1
Diagram of the coil gun circuit: an inductor ($L$) followed by a resistor ($R$) being driven by a pulse generator (PG).

The RL circuit is governed by a first-order linear differential equation:

(1) $$L\frac{dI\left(t\right)}{dt}+RI\left(t\right)=V\left(t\right),$$

where

(2) $$V\left(t\right)=\{\begin{array}{cc}{V}_0,\hfill & 0<t\le \tau \hfill \\ 0,\hfill & t>\tau .\hfill \end{array}$$

$V\left(t\right)$ is a rectangular voltage pulse provided by the pulse generator, with $V_0$ being a constant voltage and $\tau$ the pulse duration. When the pulse is on, the circuit corresponds to an RL circuit driven by a constant voltage source. When the pulse is off, the circuit behaves as a source-free RL circuit. Assuming the circuit is energized at $t=0$, and that initially the current is $I\left(0\right)=0$, solving equation 1, we find:

(3) $$I\left(t\right)=I_0\{\begin{array}{cc}1-e^{-t/{\tau }_0},\hfill & 0<t\le \tau \hfill \\ \left[\exp \left(\tau /{\tau }_0\right)-1\right]e^{-t/{\tau }_0},\hfill & t>\tau ,\hfill \end{array}$$

where ${\tau }_0=L/R$ is the circuit’s time constant, and $I_0=V_0/R$. Depending on the magnitude of ${\tau }_0$, some current could persist in the circuit after the driving voltage pulse switches off. In the limit that ${\tau }_0$ is much shorter than $\tau$ and the time scales $t$ of interest, the current pulse becomes:

(4) $$I\left(t\right)\approx I_0\{\begin{array}{cc}1,\hfill & 0<t\le \tau \hfill \\ 0,\hfill & t>\tau ,\hfill \end{array}$$

and the current in the circuit follows the rectangular temporal shape of the voltage provided by the pulse generator.

2.2. Magnetic model

When an electric current energizes the solenoid coils, a ferromagnetic projectile can be propelled by the electromagnetic force of the coils. Consider the finite solenoid shown in Figure 2. It has length $l$, radius $r$ and $N$ turns. In general, at an arbitrary point in space, a current $I$ flowing through the solenoid produces a magnetic field $\mathbf{B}$ with both radial and longitudinal components. Along the axis of the solenoid, however, the radial component vanishes. The longitudinal component can be obtained by modeling the discrete windings of the solenoid as a continuous distribution of infinitesimal current loops carrying a uniform current. Applying the Biot–Savart law to an infinitesimal current loop and integrating the resulting contributions over the length of the solenoid yields the following expression for the longitudinal component of the magnetic field: [⁶, ¹⁸]

Figure 2
Schematic diagram of the finite solenoid showing its parameters. For clarity, we show only a few coils.

(5) $$B_x=\frac{{\mu }_{0}nI}{2}\left[\frac{l/2-x}{\sqrt{{\left(l/2-x\right)}^2+r^2}}+\frac{l/2+x}{\sqrt{{\left(l/2+x\right)}^2+r^2}}\right],$$

where $x$ is the axial position, measured from the center of the solenoid; $n=N/l$ is the number of turns per unit length; and ${\mu }_0$ is the vacuum magnetic permeability. When the length of the solenoid is much larger than its radius ($l>>r$), equation (5) gives the well known infinite solenoid approximation: $B_0={\mu }_{0}nI$. Although uniform around the center of the solenoid, where it is strongest, the magnetic field near the solenoid’s ends is not homogeneous, as illustrated in Figure 3(a). The magnetic field decreases monotonically to zero, as the distance from the ends increases. This inhomogeneity of the magnetic field of the finite solenoid is crucial for launching a projectile from the solenoid, as it gives rise to a magnetic force.

Figure 3
(a) The magnetic field produced by a solenoid along its axis and (b) the magnetic force exerted on the projectile by the solenoid’s magnetic field as a function of position $x/l$, both in arbitrary units. While the force is attractive in one of the ends of the solenoid, it is repulsive at the opposite end. The magnetic field and the magnetic force are calculated for a solenoid with ratio $r/l=0.14$, similar to the solenoid we used in our experiment. In both (a) and (b), the gray vertical line marks the position ($x=0$) of the center of the solenoid.

To calculate the magnetic force acting on the projectile, we consider a spherical ferromagnetic projectile of permeability $\mu$ and radius $r_0$. Initially, before the magnetic field is turned on, the projectile is unmagnetized. As the magnetic field is turned on and penetrates into the projectile, the projectile becomes magnetized. The resultant magnetization $\mathbf{M}$ of the projectile, subjected to a magnetic field $\mathbf{B}$, is[¹⁷]

(6) $$\mathbf{M}=\left[\left(\mu /{\mu }_0\right)-1\right]\mathbf{B}/{\mu }_0.$$

The projectile’s dipole moment is $\mathbf{p}=V\mathbf{M}$, where $V=4\pi r_0^{\text{3}}/3$ is the projectile’s volume. A dipole in a magnetic field $\mathbf{B}$ has a potential energy $-\mathbf{p}\cdot \mathbf{B}$. The magnetic field acts to align the dipole moment by exerting a force given by the negative of the potential energy’s gradient: [², ¹⁷, ¹⁹]

(7) $$\mathbf{F}=\nabla \left(\mathbf{p}\cdot \mathbf{B}\right)$$

(8) $$=\left(V\left[\left(\mu /{\mu }_0\right)-1\right]/{\mu }_0\right)\nabla {\left|\mathbf{B}\right|}^2.$$

Since only the longitudinal component of the magnetic field $B_x$ exists along the axis of the solenoid, the magnitude of the magnetic force exerted on the projectile is:

(9) $$F=a\frac{d{B}_x^2}{dx},$$

where $a=\left(4\pi r_0^{\text{3}}/3{\mu }_0\right)\left(\mu /{\mu }_0-1\right)\approx 4\pi r_0^{\text{3}}\mu /3{\mu }_0^{\text{2}}$, for $\mu \gg {\mu }_0$. Figure 3(b) shows the axial magnetic force $F$. The force is largest near the ends of the solenoid and zero towards its center, where the magnetic field is more homogeneous. And, as the projectile passes the center of the solenoid, the force exerted by the magnetic field reverses direction. If the solenoid coils are driven by a current pulse, $F$ is the instantaneous force acting on the projectile for the duration of the pulse.

2.3. Mechanical model

We want here to investigate the dynamics of the projectile inside the solenoid as a function of the duration of the current pulse. We can write the equation of motion for the projectile, moving along the $x$ axis of the solenoid:

(10) $$m\frac{d^{2}x}{d{t}^2}=F+F_d,$$

where $m$ is the projectile’s mass, and $F_d$ is the dynamic sliding friction force between the solenoid’s inner wall and the spherical projectile. For a hard sphere moving on a soft surface (our solenoid was wound around a plastic tube), $F_d=-bv$, where $b$ is a parameter characterizing the surface on which the sphere is moving and the contact area, and $v$ is the projectile’s velocity [²⁰]. We neglect the force due to air drag on the sphere. Because the driving current is time dependent, the magnetic field is also a function of time $t$ as well as position $x$:

(11) $$B_x=B_x(x,t)=\frac{{\mu }_{0}n{I}_0}{2}h\left(t\right)g\left(x\right),$$

where $g\left(x\right)$ is given by the terms in square brackets in the right hand side of equation (5) and $h\left(t\right)$ by the time dependencies of equation (3). In equation 11, we assumed a quasi-static approximation, in which the magnetic field responds almost instantaneously to changes in the driving current.

The equation of motion becomes

(12) $$m\ddot{x}+b\dot{x}-ah\left(t\right)f\left(x\right)=0,$$

where $f\left(x\right)={\left({\mu }_{0}n{I}_0/2\right)}^{2}d{g}^2\left(x\right)/dx$.

Due to the complexity of the $f\left(x\right)$ function, a general analytic solution to equation 12 is unknown to us. (When the driving pulse shuts off at $t=\tau$, equation (12) simplifies and predicts a simple exponential decay of the velocity due to friction: $v\left(t\right)\propto \exp \left[-\left(b/m\right)\left(t-\tau \right)\right]$.) Therefore, we solve equation (12) numerically to find the position and velocity of the projectile inside the solenoid as a function of time for different durations of the current pulse.

2.4. Limitations of the models

To simplify the analysis, some effects were neglected in our electrical and magnetic models. In principle, the inductance of the solenoid depends on the projectile’s position inside the solenoid, which in turn affects both the current and the magnetic force acting on the projectile. Because the position of the projectile changes with time, the inductance could also become time dependent: $L=L\left(x\left(t\right)\right)$. However, we saw no experimental evidence of such a time dependence in our solenoid. Accordingly, this effect was neglected in equation 1. In general, the magnetic permeability depends on the applied magnetic field: $\mu =\mu \left(B\right)$, and for ferromagnetic materials, this dependence is highly nonlinear. At sufficiently high magnetic fields, the permeability approaches saturation. However, under the experimental conditions of the present work, the magnetic field generated by the solenoid ($\lesssim 0.004$ T) is several orders of magnitude lower than the saturation field of the projectile material ($\sim 1$ T). We treated the projectile as a soft ferromagnetic material, for which the remanent magnetization is relatively small. Under this assumption, hysteresis can be neglected, and $\mu \left(B\right)$ is a single-valued function. Nonetheless, we completely neglected this field dependence and assumed the permeability $\mu$ to be constant throughout the analysis.

Simplifications were also made in the mechanical model. The dynamics of a sphere moving on a horizontal surface under the influence of friction are not straightforward. Even if the sphere starts from rest without any angular velocity, it may eventually exhibit rolling and spinning in addition to sliding. Accurately modeling the friction associated with all three modes of motion greatly complicates the problem [²¹]. In our model, we considered only sliding friction and neglected the sphere’s rolling and spinning motions. We also neglected the effect of air drag on the sphere’s motion [²²]. These simplifications could lead to an overestimation of the sphere’s launch velocity.

A predictive numerical evaluation of the forces exerted on the projectile, and therefore of its motion, requires knowledge of the projectile’s magnetic permeability, $\mu$, and the friction constant, $b$. However, we did not measure either of these quantities and instead treated them as adjustable parameters of the model.

3. Experimental Setup

Figure 4(a) shows a picture of our experimental setup. A solenoid, 33 mm long with $350$ turns, was wound around a rigid plastic tube with insulated 20 AWG copper wire. For a constant current of 3 A, the magnetic field at the center of the solenoid is estimated to be approximately 40 Gauss. The solenoid has a mean diameter of 12.8 mm. It has a total resistance of 1.8 $\Omega$ and an inductance of 375 $\mu$H, measured with a standard multimeter, giving for the RL circuit a time constant of ${\tau }_0=0.2$ ms. The solenoid was mounted on MDF (medium-density fiberboard) support brackets for horizontal launching of the projectile. For projectile, we used a steel sphere with a 5 mm diameter and 0.7 g mass. The magnetic permeability of the sphere is unknown. To measure the exit velocity of the projectile, we placed a photogate detector¹ near the output end of the solenoid. A hole – seen in Figure 4(b) – was drilled through the plastic tube at the photogate location to allow the infrared beam from the photogate’s emitter to reach its detector without scattering from the tube wall or its graduation marks. Our photogate is home-built and has a resolution of 0.1 ms. Since the inner diameter of the solenoid tube is 5.8 mm, the projectile moves inside the solenoid with its center closely aligned with the axis of the solenoid, Figure 4(c). There is no electrical contact between the coils and the projectile. We used a fixed stop to guarantee that, for every launch, the projectile would start from the same position (approximately 4 mm from the input end of the solenoid). We did not attempt to optimize the coil gun parameters or the start position of the projectile for maximum exit velocity.

Figure 4
(a) Photograph of the experimental setup showing the coil gun (CG) mounted on MDF brackets, the photogate (PG), and the LN298N H-bridge driver. The solenoid coils were wound around a plastic tube cut from a serological pipette, which serves as a track for the projectile. The photogate was positioned near the end of the solenoid from which the projectile was launched. At the opposite end of the coil gun, a glass rod with a rounded tip (not shown) was fixed in place to serve as a mechanical stop defining the projectile’s initial position. (b) Detail of the solenoid showing the hole (indicated by the red arrow) drilled in the plastic tube. (c) Cross-sectional drawing of the solenoid and spherical projectile, including their geometric parameters. The center of the sphere lies 0.4 mm below the solenoid axis.

To generate the current pulse, we used, for convenience, a digital pulse generator (SRS DG535). (The pulses could have easily been generated with an Arduino board or a basic function generator.) The generator delivers 5 V, TTL (Transistor-Transistor Logic) pulses that only drive an average current of 0.8 mA since it is designed to drive 50 $\Omega$ loads. To effectively drive the solenoid, the TTL pulse from the generator needs to be amplified. To this end, we used a L298N H-bridge motor driver. This driver accepts standard TTL logic levels and amplifies them, for a total current up to 4 A. The driver has two power output stages, but we used only one of them. We powered the driver with a 12 V DC power supply. Figure 5 shows an schematic diagram of how the different components were connected. The DC power supply is connected to the Vcc terminal. The power supply and the pulse generator are grounded together and connected to the driver’s ground terminal. The output signal from the pulse generator is connected to the IN1 and IN2 inputs of the driver. And finally, the solenoid wires are connected to the driver’s OUT1 and OUT2 terminal. The L298N driver has fast response times, in the order of 1 $\mu$s or less. Figure 6 shows the voltage measured across the solenoid for an input TTL pulse with 45 ms duration. The voltage pulse closely follows the input TTL pulse over the time scales of interest (tens of milliseconds or longer), but with a higher power-driving capability. Those time scales are also two to three orders of magnitude longer than the time constant of the RL circuit. As seen in the inset of Figure 6, the back induced emf in the RL circuit decays quickly, within less than 0.5 ms. The current driving the coil gun can then be approximated as rectangular to a very good degree with the same duration $\tau$ as the voltage pulse [equation (4)].

Figure 5
Circuit schematics for amplification of TTL pulses using the L298N H-bridge driver. TTL pulses are provided by the Stanford Research Systems (SRS) DG535 pulse generator. The L298N driver is powered by a dc power supply (12 V, 3 A). The onboard 5V-Enable jumper in the L298N driver (green rectangle) was kept on.

Figure 6
Measured voltage drop across the solenoid terminals (solid black line). The driving TTL pulse (dashed red line) has a duration of 45 ms and an amplitude of 4 V. Except for the brief transient response associated with the solenoid’s inductance when the driving pulse is switched on and off, the voltage across the solenoid follows the TTL pulse, but at a slightly higher amplitude of 4.6 V. (In principle, the 12 V supply applied to the L298 H-bridge driver should appear at its output. However, the solenoid load tries to draw more current than the driver can deliver, causing the output voltage to saturate at approximately 4.6 V.) The inset shows an expanded view of the same signals over a much shorter time interval, highlighting the transient response of the solenoid. These traces were acquired during a single TTL pulse while the projectile was moving inside the solenoid. They demonstrate that, to a very good approximation, the solenoid’s inductance is independent of the sphere’s position within the solenoid.

We varied the pulse duration from 25 ms (the shortest duration for which the projectile exited the solenoid) up to 440 ms. The photogate registered the time during which its infrared beam was blocked by the projectile. Using the diameter of the projectile and the blocked time, we determined the velocity. For each pulse duration, we launched the projectile 15 times to reduce the statistical uncertainty in the blocked time. The main source of uncertainty in the measured blocked times is the initial position of the projectile. Although we attempted to minimize this uncertainty by positioning the projectile against a fixed stop, it tended to shift slightly after being placed in position for each launch. The resulting uncertainty in the measured blocked time is typically about 3 ms, corresponding to a velocity uncertainty of approximately 0.5 cm/s. The vertical alignment of the photogate relative to the center of the projectile can affect the measured blocked time by up to 10%. However, because the photogate remained fixed throughout the data acquisition, any misalignment contributes only to a systematic uncertainty in the blocked time.

4. Results and Discussions

We numerically solved equation 12, using MATHEMATICA’s built-in NDSolve function², to investigate the dynamics of the projectile’s motion inside the solenoid. Figure 7 shows the position and velocity of the launched projectile as a function of time for four different durations of the current pulse: (a) 15 ms, (b) 50 ms, (c) 207 ms and (d) 410 ms. In all cases, the projectile is initially at rest at one of the solenoid’s end ($x=-l/2$). In (a), while the current is on, the projectile is accelerated towards the opposite end of the solenoid. Once the current is shut off, the projectile moves under the influence of friction alone, which decelerates the projectile. The black arrow marks the moment when the projectile leaves the solenoid. In (b), the current pulse is still on when the projectile approaches the center of the solenoid, and the projectile starts to feel the effects of the repulsion force from the opposite end of the solenoid. The velocity reaches a maximum value before the projectile reaches the center of the solenoid, and then it starts to decrease while the pulse is still on. After the pulse is turned off, the projectile still moves outwards from the solenoid, with the velocity decaying exponentially with distance traveled. The projectile is ejected from the solenoid approximately 25 ms after the current pulse is shut off. If the goal is to optimize the velocity with which the projectile is launched, clearly the current pulse duration should be shorter than the 50 ms used here. But, if the pulse is too short, as in (a), the projectile will not pick up sufficient velocity before it starts moving under the influence of only the friction force. The velocity peaks at approximately 31 ms, before the projectile reaches the center of the solenoid. A pulse current of that width would optimize the launch velocity.

Figure 7
Position (dotted blue line) and velocity (solid red line) of the projectile as a function of time inside the solenoid for four current pulse durations: (a) 15 ms, (b) 50 ms, (c) 207 ms and (d) 410 ms (without friction). In all cases, the projectile starts at $x=-l/2=-1.65$ cm, and the black arrow points to the moment when the projectile leaves the solenoid, at $x=+l/2$. The green dashed line is the current pulse. The dotted horizontal lines are the position of the ends of the solenoid ($x=\pm l/2)$. The solenoid’s numerical parameters used in the simulation correspond to those of the experiment. We also used ${\left({\mu }_{0}n{I}_0/2\right)}^{2}a=1.1\times {10}^{-4}$ kg$\cdot$m/s², and $b=2.9\times {10}^{-3}$ kg/s in solving equation 12.

Figure 7(c) shows the case in which the pulse duration is 207 ms. The pulse duration is long enough that the projectile gets close to the $x=+l/2$ end of the solenoid, it is repelled by the magnetic force at that end and sent back towards its initial position. Before the projectile arrives at $x=-l/2$, it is repelled by the magnetic force at that end of the solenoid, reversing direction one more time. It is again accelerated towards the output end at $x=+l/2$. The current pulse is turned off at the point of maximum velocity. From there on, the projectile is decelerated by friction. Because of friction, the maximum velocity reached the second time around the solenoid is smaller than that reached in the first round. Curiously, had the current pulse been about half as short in duration, the projectile would have been launched out of the solenoid from the same end it started from ($x=-l/2$), moving in the negative $x$ direction. In the absence of friction, [Figure 7], and within the framework of the model, in which no other mechanisms for energy loss are included, the projectile would oscillate back and forth between the two ends of the solenoid for as long as the current pulse remained on.

In our experimental setup, lateral optical access to the interior of the solenoid was not available, preventing direct side-view observation of the projectile’s motion. However, the Supplementary Material includes three videos recorded from a frontal view of the solenoid. Video 1 shows the projectile’s motion for a current pulse duration of 70 ms. In this case, the projectile undergoes a simple motion in one direction, from the rear end toward the front end of the solenoid. In Video 2, corresponding to a pulse duration of 240 ms, the projectile is seen approaching the front end of the solenoid and then reversing its direction of motion. After returning to the rear end, it moves toward the front once again and ultimately exits the solenoid. In Video 3, for a pulse duration of 410 ms, the projectile completes two round trips within the solenoid before finally moving toward the front end and exiting the device. The results of the theoretical model shown in Figure 7 are in agreement with the dynamics seen in the videos.

In Figure 8, we present the experimental results for the projectile’s exit velocity as a function of the current pulse duration, together with the predictions of the theoretical model. The model shows excellent qualitative agreement with the data and accurately reproduces the overall behavior of the system. However, it systematically overestimates the exit velocities. We attribute this discrepancy to the fact that the model neglects the projectile’s spinning and rolling motions and considers only its translational motion. The adjustable parameters used in the model, obtained by manual fitting of the data, are presented in Table 1. Both model and data exhibit three velocity maxima, each successively lower than the preceding one, due to frictional losses. For pulse durations in the vicinity of the first maximum, the projectile undergoes a single pass through the solenoid, simply moving from the rear to the front end (as seen in Video 1). For pulse durations near the second maximum, the projectile moves back and forth once inside the solenoid before exiting (Video 2). Finally, for pulse durations around the third maximum, the projectile completes two round trips within the solenoid before being launched (Video 3). No data points are shown between these three sets of pulse durations because, in those intervals, the projectile did not exit the solenoid.

Figure 8
Exit velocity of the projectile as a function of pulse duration. When multiplied by an arbitrary scaling factor of 0.6, the model agrees remarkably well with the experimental data. This indicates that, although the model accurately captures the qualitative behavior of the system, it systematically overestimates the projectile’s exit velocity. Uncertainty bars are one standard deviation.

Table 1
Values of the adjustable parameters used in the model. For a current of 3 A, the parameter listed in the first row gives a relative magnetic permeability $\mu /{\mu }_0\approx 250$, which lies within the expected range for carbon steel.

Parameters	Value
${\left({\mu }_{0}n{I}_0/2\right)}^{2}a$	$5\times {10}^{-5}$ $\text{kg}\cdot {\text{m/s}}^2$
$b$	$2.6\times {10}^{-3}$ kg/s
$x\left(0\right)$	$-2.09$ cm

The velocities reported here are substantially lower than those typically observed in coil gun systems [⁶]. This difference can be attributed to the fact that our projectile is not permanently magnetized, the driving currents are low, and the solenoid has a relatively high diameter-to-length ratio. Together, these factors result in a comparatively weak magnetic force and, consequently, low launch velocities.

5. Conclusions

In conclusion, we presented an experiment designed to investigate the dynamics of a metallic projectile inside a coil gun as a function of the duration of the current pulse driving the solenoid. We demonstrated that, for sufficiently long current pulses, the projectile exhibits oscillatory motion, moving back and forth between the ends of the solenoid, before being launched. For our particular coil gun design, an optimal pulse duration of approximately 70 ms was found to maximize the projectile’s exit velocity. Two additional local maxima in the launch velocity were observed at pulse durations of 235 ms and 414 ms. Although the proposed model is based on several simplifying assumptions, it successfully captures the main features observed in the experiment and exhibits excellent qualitative agreement with the experimental data. However, it systematically overestimates the launch velocities. The model can be used to quickly estimate the optimal current pulse duration for a given coil gun apparatus.

The experiment, including both its underlying physics and its apparatus, is suitable for undergraduate teaching laboratories. In fact, it was conducted by the authors as part of a one-semester, project-based physics laboratory course for undergraduate STEM students. The cost and simplicity of the setup are comparable to those of traditional coil gun experiments intended for instructional use. Although an expensive pulse generator was employed for convenience, as it was readily available, the apparatus could be made substantially more affordable and accessible by replacing the pulse generator with an Arduino board, for example. The setup is safe for use in a teaching laboratory: it does not require a high-voltage/high-current power supply, nor does it rely on high-capacitance capacitors. The projectile’s exit velocities are low, on the order of tens of cm/s, and do not pose a safety risk to students. As in many physics problems, the equation of motion for the projectile does not appear to admit an analytic solution. Involving students in modeling the experiment provides them with a valuable opportunity to explore numerical simulation techniques.

Finally, some of the results discussed here – particularly the observation of pulse durations that maximize the projectile’s exit velocity – may be relevant to the design of more advanced coil gun systems.

Supplementary Material

The following online material is available for this article:

Video 1 – Frontal view of the coil gun showing the projectile’s movement during a 70 ms current pulse. The projectile travels along the solenoid axis, moving from the rear end toward the front end and exiting through the front.

Video 2 – Frontal view of the coil gun showing the projectile’s movement during a 240 ms current pulse. The projectile travels toward the front end of the solenoid, where it is pushed back toward the rear end. At the rear end, it reverses direction once more and travels toward the front end until it exits the solenoid.

Video 3 – Frontal view of the coil gun showing the projectile’s movement during a 410 ms current pulse. The projectile travels back and forth between the rear and front ends of the solenoid twice before finally reversing direction and exiting through the front end.

Data Availability

The entire dataset supporting the results of this study is published in the article.

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¹

A photogate is a precision timing instrument commonly used in undergraduate physics labs. It consists of two arms separated by a narrow gap. One arm houses an infrared light emitter, while the opposite arm contains a sensor that continuously detects the emitted beam. When an object passes through the gap and interrupts the beam, the sensor detects the interruption and records the event. The photogate measures the time between the moment the beam is blocked and the moment it is restored. If the length of the object passing through the gate is known, its speed can be calculated by dividing that length by the measured blocked time. [²³, ²⁴]

²

The user can specify the numerical method that NDSolve uses to solve the differential equation, but we allowed it to select the most appropriate method automatically. In this case, it seamlessly switches between Adams methods for non-stiff systems and Gear (Backward Differentiation Formula, BDF) methods for stiff systems.

*Correspondence email address:

[e-mail]

Editor-in-Chief:

Marcello Ferreira https://orcid.org/0000-0003-4945-3169

Publication Dates

• Publication in this collection
  02 Nov 2026
• Date of issue
  2026

History

• Received
  30 Mar 2026
• Reviewed
  04 Aug 2026
• Accepted
  19 Aug 2026