AP Physics 1 · Lesson 1

Momentum and impulse

Know Know it

Momentum is mass times velocity, p=mvp = mv, in kg·m/s. Because velocity has a direction, so does momentum: choose a positive direction and give every momentum a sign.

A force acting for a time gives an object an impulse, J=FΔtJ = F\Delta t, in N·s. The impulse-momentum theorem says the net impulse equals the change in momentum: FnetΔt=Δp=m(vf−vi)F_{\text{net}}\Delta t = \Delta p = m(v_f - v_i). The units match: one N·s is one kg·m/s.

When the force changes during the contact, the impulse is the area under the force-time graph. Split the area into triangles and rectangles, and count any area below the time axis as negative.

Three ways to find an impulse
What you are givenHow to get the impulse
A constant (or average) force and a timeJ=FΔtJ = F\Delta t
A graph of force against timeThe area between the graph and the time axis
The mass and the velocities before and afterJ=Δp=m(vf−vi)J = \Delta p = m(v_f - v_i), with signs

Calculator On the exam

On the exam you may use a four-function, scientific or graphing calculator on both sections, and the testing app also has built-in scientific and graphing calculators. You get an equation sheet with reference tables too, so your job is choosing the right relationship and setting it up, not memorizing it. Keep track of milliseconds: 1 ms is 0.001 s.

Worked example Easy

A 19 kg curling stone slides across the ice at 2.5 m/s. What is the magnitude of its momentum?

  1. 7.6 kg·m/s
  2. 47.5 kg·m/sAnswer
  3. 59.4 kg·m/s
  4. 119 kg·m/s

How to solve it

  1. Momentum is mass times velocity: p=mvp = mv.
  2. p=(19 kg)(2.5 m/s)=47.5p = (19 \text{ kg})(2.5 \text{ m/s}) = 47.5 kg·m/s, in the direction the stone slides.

Why each choice is right or wrong

  • A. Incorrect. 7.6 kg·m/s divides the mass by the speed; momentum is the product of mass and velocity.
  • B. Correct. p=mv=(19)(2.5)=47.5p = mv = (19)(2.5) = 47.5 kg·m/s.
  • C. Incorrect. 59.4 kg·m/s is 12mv2\frac{1}{2}mv^2, the kinetic energy in joules, not the momentum.
  • D. Incorrect. 119 kg·m/s is mv2mv^2, which squares the speed; momentum uses the speed once.

Trap A bounce changes momentum more than a stop

Write Δp=m(vf−vi)\Delta p = m(v_f - v_i) with signs. An object that stops changes its momentum by mvmv; one that bounces straight back at the same speed changes it by 2mv2mv. Subtracting speeds without signs makes a bounce look like a small change, or no change at all.

Explain Explain it

Task verbs: calculate, justify

Rearranged, the theorem gives the average force during a contact: Favg=Δp/ΔtF_{\text{avg}} = \Delta p/\Delta t. For the same change in momentum, stretching out the contact time lowers the average force. That is why padding, bent knees and airbags protect you: they do not change how much momentum you lose, only how long it takes.

The direction of the impulse is the direction of the change in momentum, not the direction of motion. A ball moving right that is hit back to the left receives an impulse to the left.

Worked example Medium

A 0.15 kg baseball moving at 38 m/s toward a batter leaves the bat at 45 m/s in the opposite direction. The bat touches the ball for 1.2 ms. What is the magnitude of the average force of the bat on the ball?

  1. 10.4 N
  2. 875 N
  3. 5,630 N
  4. 10,400 NAnswer

How to solve it

  1. Take the direction the ball leaves as positive: vi=−38v_i = -38 m/s and vf=+45v_f = +45 m/s.
  2. Δp=m(vf−vi)=(0.15)(45+38)=12.45\Delta p = m(v_f - v_i) = (0.15)(45 + 38) = 12.45 kg·m/s.
  3. Favg=Δp/Δt=12.45/0.0012≈10,400F_{\text{avg}} = \Delta p/\Delta t = 12.45/0.0012 \approx 10{,}400 N.

Why each choice is right or wrong

  • A. Incorrect. 10.4 N divides by 1.2 instead of by 0.0012 s; the time must be in seconds.
  • B. Incorrect. 875 N subtracts the speeds, 45 - 38; the ball reverses, so the changes add.
  • C. Incorrect. 5,630 N uses only the final momentum and ignores the momentum the ball had coming in.
  • D. Correct. Δp=(0.15)(45−(−38))=12.45\Delta p = (0.15)(45 - (-38)) = 12.45 kg·m/s, and 12.45/0.0012≈10,40012.45/0.0012 \approx 10{,}400 N.

Justify Your turn

A climber falls and is caught by her rope. Climbing ropes are built to stretch. Explain why a stretchy rope is safer for her than a steel cable that would stop her from the same speed.

Write it, then compare. Nothing you type here is saved or graded.

Compare with a model answer

A full answer does this

  • Says her change in momentum (the impulse needed to stop her) is the same with either rope
  • Says the stretchy rope stops her over a longer time
  • Says the average force on her is smaller with the stretchy rope

Model answer

Either way she goes from the same speed to rest, so her change in momentum, and the impulse that must act on her, is the same. The stretchy rope spreads that impulse over a longer stopping time. Since the average force equals the change in momentum divided by the time, the longer time means a smaller average force on her body, which is what makes the fall less dangerous.

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