1. Real‑world motivation

Impulse describes how a force acting over time alters an object's momentum. When a bat strikes a baseball or a thruster nudges a satellite, what matters is not merely the force but the integral of force with respect to time. Traditional calculators multiply force and duration to return a number in newton‑seconds, yet they provide no sense for how the momentum builds. This simulator transforms the impulse–momentum theorem into an interactive experience. As the orange force arrow pushes the blue puck, striped bars underneath show impulse and momentum change growing in lockstep. The caption narrates the instant time, velocity, impulse, and momentum so screen‑reader users follow the same story. Watching the puck coast once the force shuts off reveals why modern crash systems stretch impact time—to deliver the same momentum change with a gentler force.

2. Variables and assumptions

The simulation considers a point mass $m$ moving horizontally on a frictionless surface. At time zero the object has velocity $v_0$. A constant force $F$ acts in the positive direction for a specified duration $t_F$. After the force interval ends, the object coasts with whatever velocity it has attained. Inputs are constrained to SI units: kilograms, newtons, seconds, and meters per second. The time step $\mathrm{\Delta t}$ is clamped between 0.001 and 0.1 s and the total simulation time must exceed the force duration. Neglecting friction, air resistance, and rotational effects simplifies the dynamics to one dimension while preserving the essence of impulse.

3. Governing equations

The impulse–momentum theorem states $J=\Delta p$, where $J$ is impulse and $p$ is linear momentum $mv$. For a constant force the impulse delivered up to time $t$ is $\mathrm{J(t)}=F\backslash min(t,t_F)$. The acceleration while the force is active is $a=\frac{F}{m}$, leading to velocity $\mathrm{v(t)}=v_0+at$ for $t\le t_F$ and constant thereafter. The momentum change is $\Delta p=m(\; msup\{v\}\{\; \}-\; msub\{v\}\{0\})$. The simulator computes both quantities each frame and displays their difference $\mathrm{\Delta E}$ to highlight numerical error.

4. Numerical scheme

An explicit Euler method suffices for these first‑order equations. At each time step the algorithm determines whether the force is still active. If $t<t_F$, the acceleration is $\mathrm{F/m}$; otherwise it is zero. Velocity and position update via $\mathrm{v\_\{n+1\}}=\mathrm{v\_n}+a\mathrm{\Delta t}$ and $\mathrm{x\_\{n+1\}}=\mathrm{x\_n}+\mathrm{v\_\{n+1\}}\mathrm{\Delta t}$. Impulse accumulates as $\mathrm{J\_\{n+1\}}=\mathrm{J\_n}+F\mathrm{\Delta t}$ during force application. Because acceleration is piecewise constant, the scheme is numerically stable for the constrained step sizes; any drift between impulse and momentum is reported as $\mathrm{\Delta E}$ in the caption.

5. Worked example

Imagine a 0.5 kg puck sliding on ice at $v_0=1$ m/s. A 2 N shove acts for 0.5 s. The analytic impulse is $J=F{t}_F=1$ N·s. The resulting momentum change is also 1 N·s, so the final velocity should be 3 m/s. Running the simulator with $\mathrm{\Delta t}=0.01$ s shows the puck accelerating while the force arrow is visible. After half a second the arrow disappears, the puck continues at constant 3 m/s, and the impulse and momentum bars freeze at equal lengths. Exporting the CSV confirms the numbers: at 0.5 s the file lists $J=1.00$ and $\Delta p=1.00$ within machine precision.

6. Comparison table

The table compares how varying mass or force affects the final state for a fixed 1 s force duration.

m (kg)	F (N)	Final v (m/s)	Impulse (N·s)
1	10	10	10
2	10	5	10
1	5	5	5

Doubling mass halves the velocity change while impulse stays fixed. Halving force halves both impulse and final speed. The simulation makes these relationships tangible as the bars and motion respond instantly.

7. How to read the animation

A blue puck glides along the canvas. During the force interval, an orange arrow pushes it to the right and the puck accelerates, leaving a trail. Beneath the canvas, the orange bar labelled “Impulse” lengthens linearly with time while the blue “Momentum change” bar mirrors it; their matching lengths confirm the theorem. Once the force ceases, the arrow disappears and both bars freeze, while the puck coasts at constant speed. The caption and hidden text summarize time, velocity, impulse, and momentum so that all users, including those relying on screen readers, gain the same insight. Keyboard users can focus the canvas and toggle play and pause with the space bar.

8. Limitations

The model assumes a perfectly rigid object on a frictionless surface with a strictly constant force. Real collisions involve forces that vary rapidly, deformation, and energy losses. When forces act at angles or over two dimensions, vector components must be considered. The simulation ignores rotational impulse and treats momentum as a scalar, adequate only for straight‑line motion. Large time steps may cause the impulse and momentum bars to diverge slightly; reducing $\mathrm{\Delta t}$ mitigates the discrepancy.

9. Suggested extensions

Enhancements could include custom force–time profiles, allowing users to sketch arbitrary curves and see the resulting momentum. Coupling the puck to another mass would demonstrate conservation of momentum in collisions, complementing our Elastic Collision Simulator and Inelastic Collision Simulator. Adding friction would connect the tool with the Work from Force with Friction page. A phase‑space plot of momentum versus position would further illustrate the buildup of motion.

10. References

For deeper study consult J. R. Taylor, Classical Mechanics, which derives the impulse–momentum theorem from Newton's laws. R. A. Serway and J. W. Jewett's Physics for Scientists and Engineers provide numerous example problems. Discussions of real collision forces can be found in journals of biomechanics and materials science. NASA's educational resources also explain how spacecraft rely on impulsive thruster firings for navigation.