Physics · Mechanics

Momentum and Impulse

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What you learned

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Read next: Conservation of MomentumCarry impulse into multi-object motion

Key takeaway

Momentum combines mass, speed, and direction.
Impulse is the signed area under the force-time graph.
Equal impulse gives the same momentum change even when the force pulse has a different shape.
A heavier cart can get the same momentum change with a smaller change in speed.

Common misconception

A bigger force automatically means a bigger final momentum change, even if the pulse is much shorter.

Momentum change depends on total impulse, which is the signed force-time area, not on force size alone. A large force over a short time can match a smaller force acting longer.

Keep force and motion connected

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Two-object systemsMechanics

Conservation of Momentum

Use two carts to see the conservation rule in action: with no outside push, the system's total momentum stays fixed even while each cart's momentum and speed change.

Collision outcomesMechanics

Collisions

Collide two carts on one honest track, keep total momentum in view, and see how elasticity, mass, and incoming speed shape the rebound or stick-together outcome.

Apply it to trajectoriesMechanics

Projectile Motion

Launch a projectile, watch the trajectory form, and connect the range, height, and component motion to the launch settings.

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Equation snapshotMomentum · Impulse for a constant pulseShow 3 equations

Momentum
$p = m v$
Momentum combines mass with velocity, so direction matters as well as size.
Impulse for a constant pulse
$J = F Δ t$
For a constant force, impulse is the signed area under the force-time graph.
Impulse-momentum theorem
$J = Δ p = p_{f} - p_{i}$
Impulse changes momentum directly, which is why the accumulated impulse and the momentum change should match.

Why it behaves this way

Explanation

Momentum measures motion with direction: in one dimension, $p = m v$ . A heavier cart or a faster cart has more momentum, and reversing direction changes the sign of the momentum.

Impulse is the total effect of a force acting over time. In this module, one bounded force pulse acts on one cart, so you can compare the force graph, the accumulated impulse, and the momentum graph directly. The key idea is that the same signed force-time area produces the same change in momentum.

Key ideas

01Momentum depends on mass, speed, and direction, so the same speed change does not mean the same momentum change for every cart.

02Impulse is the signed area under the force-time graph. A taller, shorter pulse and a lower, longer pulse can produce the same impulse if their areas match.

03For the same cart, impulse changes momentum directly through

J = Δ p

, so the accumulated-impulse graph and the momentum change should match.

Worked examples

Solve the live pulse

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Frozen walkthrough

Step through the frozen example

Frozen walkthrough

Use the current pulse as evidence. First find the cart's momentum at the instant shown. Then use the same force and pulse duration to predict the impulse and final momentum, and compare that prediction with the graphs and the stage.

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Example 1 of 2

Frozen valuesFrozen at 0.00

At $t = 0 s$ , what is the cart's momentum?

t

Time

0 s

m

Mass

1 kg

v

Cart velocity

0.5 m/s

1. Write the momentum relation

Use

p = m v

for the cart at the currently inspected moment.

2. Substitute the live values

p = 1 \cdot 0.5

3. Calculate the momentum

That gives

p = 0.5 kg m/s

Current momentum

p = 0.5 kg m/s

Before the pulse starts, the cart keeps its initial velocity, so the current momentum still matches the starting value.

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