AP® Physics 1: Algebra-Based review sheet from Aim for Five (aimforfive.com/physics/units/4/4-1)
Unit 4 · Topic 4.1
4.1 Linear Momentum
Linear momentum is mass times velocity, p = mv. It's a vector that points the same way as the velocity, and it's the main tool for analyzing collisions and explosions, where huge forces act for a short time and only the before and after states matter.
Key terms
- linear momentum
- vector
- kg·m/s
- collision
- explosion
What momentum is
Linear momentum is p = mv. Its unit is kg·m/s. Since mass is always positive, momentum points in the same direction as the velocity. In one dimension, give it the same sign as the velocity.
Momentum captures how hard it is to stop something. A slow truck and a fast baseball can have very different masses and speeds but similar momentum. In this course, "momentum" always means linear momentum unless a problem says angular momentum, which is Unit 6.
Momentum of a system
The total momentum of a system is the vector sum of its parts' momenta: p_total = m₁v₁ + m₂v₂ + …. In one dimension, add signed values. Two carts moving toward each other with equal and opposite momenta have a total momentum of zero, even though both are moving.
Dividing total momentum by total mass gives the velocity of the system's center of mass: v_cm = Σmv/Σm (4.3).
Momentum versus kinetic energy
Both depend on mass and speed, but they behave differently:
| Property | Momentum p = mv | Kinetic energy K = ½mv² |
|---|---|---|
| Type | vector | scalar |
| Can be negative? | yes (in 1D, by direction) | never |
| Double the speed | ×2 | ×4 |
| System total can be zero while parts move? | yes | no |
| Conserved in all collisions? | yes, if no net external force | only in elastic collisions |
Collisions and explosions as models
A collision is an interaction in which the forces between the objects are much larger than any outside forces during the brief contact. Friction or gravity hardly matters during those milliseconds, so you can treat the system as isolated just before and just after.
Since you only compare the moments right before and right after, you can use the object model: you don't need to know how the objects deform during contact.
An explosion is the reverse: forces inside the system push its parts apart. A firecracker bursting, a gun recoiling, or two skaters pushing off each other are all explosions in this sense.
Same momentum, different energy
Two objects can have equal momentum but different kinetic energies. Since K = p²/(2m), for the same momentum the lighter object has more kinetic energy. A bullet and a rifle recoiling from each other have equal and opposite momenta, but the bullet carries almost all the kinetic energy.
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Car versus truck
A 1200 kg car moves at 15 m/s and a 3000 kg truck moves at 6.0 m/s in the same direction. Compare their momenta and their kinetic energies.
Show the solutionHide the solution
- Step 1: Car: p = (1200)(15) = 18,000 kg·m/s. Truck: p = (3000)(6.0) = 18,000 kg·m/s. They're equal.
- Step 2: Car: K = ½(1200)(15)² = 135,000 J. Truck: K = ½(3000)(6.0)² = 54,000 J.
- Step 3: Equal momentum, but the lighter, faster car has 2.5 times the kinetic energy.
Answer: Both have 1.8 × 10⁴ kg·m/s; the car has 1.35 × 10⁵ J and the truck 5.4 × 10⁴ J
- Example 2Calculator allowed
Total momentum with opposite directions (classic trap)
A 2.0 kg cart moves right at 3.0 m/s and a 3.0 kg cart moves left at 2.0 m/s. Taking right as positive, find the system's total momentum and total kinetic energy.
Show the solutionHide the solution
- Step 1: p₁ = (2.0)(+3.0) = +6.0 kg·m/s. p₂ = (3.0)(−2.0) = −6.0 kg·m/s.
- Step 2: p_total = +6.0 + (−6.0) = 0. The center of mass isn't moving.
- Step 3: K_total = ½(2.0)(3.0)² + ½(3.0)(2.0)² = 9.0 + 6.0 = 15 J.
- Step 4: The trap is concluding there's no energy because there's no momentum. Kinetic energy has no direction, so it can't cancel.
Answer: Total momentum 0; total kinetic energy 15 J
Common mistakes
- Forgetting that momentum is a vector. Opposite directions need opposite signs, and momenta can cancel.
- Assuming equal momentum means equal kinetic energy. For the same momentum, the lighter object has more kinetic energy.
- Using the collision model when outside forces are large compared with the forces between the objects, or over a long time interval.
- Leaving out units or using g instead of kg. Momentum is in kg·m/s, which is the same as N·s.
On the exam
- Expect comparison questions: which object has more momentum, and which has more kinetic energy? Calculate both, or use K = p²/(2m).
- The course lists momentum charts among its representations, so be ready to draw or read one: a signed bar for each object's momentum before and after. When momentum is conserved, the bars must add to the same total before and after.
Connected topics
Videos
Check yourself
4 questions on 4.1 Linear Momentum. Pick an answer to see if you got it, and why.
A 0.15 kg baseball moves toward home plate at 40 m/s. What is the magnitude of its momentum?
Cart X has a mass of 2.0 kg and moves at 3.0 m/s. Cart Y has a mass of 1.0 kg and moves at 6.0 m/s. How do their momenta and kinetic energies compare?
A 2.0 kg cart moves east at 3.0 m/s, and a 1.0 kg cart moves west at 4.0 m/s. What is the total momentum of the two carts?
A 4.0 kg object has 50 J of kinetic energy. What is the magnitude of its momentum?
0 of 4 answered