Constant Acceleration Cart
Introduction
Constant acceleration is the first non-trivial motion a physics student meets. With acceleration held to a fixed value, velocity grows as a straight line in time and position grows as a parabola — two simple curves that together describe everything from a dropped stone to a car merging onto a highway. The simulator on this page treats a cart on a frictionless horizontal track as a point mass, applies the chosen acceleration starting from the chosen initial velocity, and plots position, velocity, and acceleration side by side so the two kinematic equations become visible at every frame.
The topic anchors the kinematics-to-dynamics boundary because constant acceleration is what F = m·a produces whenever the net force is constant. Once a learner can predict where the cart will be at t = 3 s before pressing Start, the two equations stop being formulas to memorise and start being statements about how the world moves. The simulator's MAX_TIME of 10 s is long enough for the quadratic in position to develop visible curvature even at modest accelerations, and short enough that a learner can hold both readouts in working memory while comparing to a hand calculation.
A common first guess is that doubling the acceleration doubles the final position. The simulator shows otherwise: with v₀ = 0 m/s, doubling a from 1 m/s² to 2 m/s² doubles the velocity at every t, but doubles the position too because the ½·a·t² term is linear in a. The non-linearity hides one level deeper — doubling the run TIME at fixed a quadruples the position, because the t² term is genuinely non-linear in t. Both relationships are visible by pausing the sim at different times and comparing the Position readout against the prediction.
The Physics Explained
The simulator integrates the two coupled equations x(t) = x₀ + v₀·t + ½·a·t² and v(t) = v₀ + a·t from t = 0 to t = 10 s. At every frame it reads the Acceleration slider (range −5 to 5 m/s², step 0.5) and the Initial Velocity slider (range 0 to 10 m/s, step 0.5), evaluates the closed-form position and velocity from the physics module, and writes them to the four readouts: Time, Position, Velocity, Acceleration. With a = 2 m/s² and v₀ = 0 m/s at t = 3 s the readouts show Time = 3.00 s, Position = 9.00 m, Velocity = 6.00 m/s, Acceleration = 2.00 m/s² — the exact analytical values, because the closed form is exact for constant a.
The left panel makes the geometry of constant acceleration visible directly. The blue cart slides rightward along a horizontal track; a blue dotted arrow above the cart points in the direction of velocity and grows in length as v grows; a red dotted arrow points in the direction of acceleration and stays constant in length because a is constant. With the default a = 2 m/s² the red arrow is fixed at a moderate length while the blue arrow starts at zero length and lengthens linearly with time — the two arrows together encode the entire kinematic state of the cart.
The right panel shows the three time-series globally instead of locally. The amber curve is x(t), the blue curve is v(t), and the red horizontal line is a(t). With v₀ = 0 m/s and a = 2 m/s² the amber x(t) is the parabola ½·a·t² that opens upward; the blue v(t) is the straight line a·t that climbs at slope 2; the red a(t) is a flat horizontal at 2 m/s². The three shapes — parabola, line, constant — are the visual signature of constant acceleration, and they remain in that hierarchy regardless of which slider values are chosen.
Setting a negative acceleration with a positive initial velocity gives the deceleration regime. With v₀ = 8 m/s and a = −2 m/s², the velocity readout decreases linearly from 8.00 m/s and crosses zero at t = 4.00 s; the position readout climbs as a parabola, peaks at 16.00 m at the moment v = 0, then declines as the cart reverses direction. This is the same kinematics a car uses when braking to a stop: the stopping distance v₀²/(2·|a|) is purely determined by the initial speed and the deceleration magnitude, and the simulator's readouts confirm it within rounding.
Key Equations
The amber curve on the right panel. With x₀ = 0 m, v₀ = 0 m/s, a = 2 m/s², and t = 3 s this evaluates to x = 0 + 0·3 + ½·2·9 = 9.00 m, exactly the value the Position readout displays when the simulator is paused at that moment. The three terms separate the three effects on position: starting location, initial drift, and acceleration-driven growth.
The blue curve on the right panel. With v₀ = 0 m/s, a = 2 m/s², and t = 3 s this evaluates to v = 0 + 2·3 = 6.00 m/s, exactly the value the Velocity readout displays at that moment. The line's slope is the acceleration; doubling a doubles the slope; reversing the sign of a flips the line's tilt and produces deceleration.
Useful when time is unknown but distance is. Setting v = 0 and solving for x gives the stopping distance under deceleration: x_stop = v₀² / (2·|a|). For v₀ = 8 m/s and a = −2 m/s² this gives x_stop = 64 / 4 = 16.00 m, exactly the peak position the sim displays at t = 4 s in that configuration. The identity is what every car's anti-lock braking system implicitly inverts to decide when to release pressure.
Under constant acceleration ONLY, the average velocity over any interval equals the arithmetic mean of the start and end velocities. For v₀ = 0 m/s and v = 6 m/s at t = 3 s, the average is 3.00 m/s and the distance covered is 3·3 = 9.00 m — exactly the Position readout. This identity is what makes constant-acceleration problems solvable by hand: the ½ in x = ½·a·t² is precisely the (v₀ + v)/2 factor pulled out, with v = a·t substituted in.
Key Variables
| Symbol | Name | Unit | Meaning |
|---|---|---|---|
| x(t) | Position | m | Cart's location along the track at time t; starts at 0 m by convention |
| v(t) | Velocity | m/s | Cart's signed speed at time t; positive is rightward |
| a | Acceleration | m/s² | Slider value, range −5 to 5 m/s², step 0.5; held constant during a run |
| v₀ | Initial velocity | m/s | Slider value, range 0 to 10 m/s, step 0.5; the velocity at t = 0 |
| t | Time | s | Elapsed time from Start; capped at 10 s by the sim's natural stop |
Real World Examples
Why does a drag racer's velocity grow as a straight line on the speedometer while distance grows as a curve?
A drag racer that holds a roughly constant acceleration during its 1320-foot run is, to first approximation, the system the simulator models: v(t) climbs linearly with the equation v = v₀ + a·t while x(t) climbs as a quadratic with x = v₀·t + ½·a·t². On the dashboard speedometer a driver sees the velocity number tick up by the same amount every second — that's the linear law. On the GPS distance readout the number does NOT tick up by the same amount every second; the second mile takes less time than the first because the car is faster on average.
The simulator makes this visible: set Acceleration to 5 m/s² and Initial Velocity to 0 m/s, run the sim, and at t = 2 s the position readout shows 10.00 m while at t = 4 s it shows 40.00 m — four times further in twice the time, exactly because the quadratic curve has the cart spending twice as long at the higher speeds it reaches in the second half of the run. The same shape governs every uniformly accelerating system: a falling stone, a launched rocket in its constant-thrust phase, an electron in a uniform electric field. Linear velocity, quadratic position — visible directly in the side-by-side plots.
How does a self-driving car decide when to start braking to stop at a red light?
A self-driving car computes its required deceleration from the same kinematic identity the simulator implements. Given a current velocity v₀ and a target velocity of 0 m/s, the stopping distance under constant deceleration a is x_stop = v₀² / (2·|a|), derived from v² = v₀² + 2·a·x with v set to 0. The simulator demonstrates this case directly: set Initial Velocity to 8 m/s and Acceleration to −2 m/s², run the sim, and the velocity readout reaches 0.00 m/s at t = 4.00 s while the position readout peaks at 16.00 m.
The car's planner runs the inverse computation every few milliseconds — given current speed and an upcoming red light at distance d, what acceleration is needed to bring v to 0 over distance d — and applies the brake force that produces that acceleration. The simulator's prediction-vs-readout agreement is the same agreement an autonomous vehicle relies on when it trusts that the brake command will produce the planned stop. A real vehicle adds road grip, brake fade, and downhill grade as second-order corrections, but the kinematic backbone is the v² = v₀² + 2·a·x identity made operational.
Why does a roller coaster designer care more about acceleration than about speed?
Roller coasters routinely reach speeds of 30 m/s or more, but the rider's body responds to acceleration, not speed — sustained acceleration of about 30 m/s² (roughly 3g) for more than a few seconds is the threshold where a healthy rider can grey-out from blood draining away from the head. The simulator's two readouts make the distinction concrete: the Velocity readout grows linearly while Acceleration stays constant at whatever the slider was set to. A coaster designer who set the slider to a = 5 m/s² and let the sim run would see velocity climb to 50 m/s by t = 10 s — fast, but only half a g of force on the rider, comfortable for the whole duration.
Setting the slider to a value that does not exist on the simulator's range (say 30 m/s²) is exactly the regime the designer refuses to enter; the simulator caps acceleration at 5 m/s² for the same reason a coaster's loop section is geometry-limited to keep the rider's experienced acceleration within survivable bounds. The takeaway is that the linear v(t) line tells you the speedometer reading; the flat a(t) line tells you what the rider FEELS. A coaster designer reads the second plot more carefully than the first.
Further Reading
- 1D motion plotter — the constant-velocity case (a = 0 m/s²), where x(t) is a straight line and the quadratic term in this article's equation vanishes.
- Velocity-time plotter — the inverse view: given a v(t) profile, reconstruct x(t) as the integral. Constant-acceleration cart is the special case where v(t) is straight.
- Average vs instantaneous velocity — the calculus limit underlying v = v₀ + a·t; for constant a the average equals the midpoint instantaneous, an identity used in this article's worked examples.
- Free fall — the canonical constant-acceleration system with a = −9.8 m/s², beyond this sim's slider range but governed by the same two equations.