Momentum is a quantity representing the “strength of motion.” When a force acts over time, momentum changes as an impulse.
Basic Knowledge
Momentum: p = mv (m: mass [kg], v: velocity [m/s], unit: kg·m/s)
Momentum is a vector quantity.
Impulse: I = FΔt (F: force [N], Δt: time of action [s], unit: N·s = kg·m/s)
Relationship between momentum and impulse (momentum theorem):FΔt = mv2 − mv1 = Δp
Impulse = change in momentum
This is obtained by integrating F = ma with respect to time: F = ma = m(Δv/Δt) → FΔt = mΔv = Δp
The area under an F-t graph represents impulse.
Momentum: p = mv [kg·m/s]
Impulse: I = FΔt [N·s = kg·m/s]
Momentum theorem: FΔt = mv2 − mv1
F-t graph: area = impulse
Direction: Momentum and impulse are both vectors. Pay attention to signs
Further Study (Background and Significance)
The momentum theorem is powerful for dealing with impact forces (when a large force acts for a short time). Since the force changes rapidly during a collision, the change in momentum can be calculated as impulse (FΔt) even when the exact change in force over time is unknown.
The principle of airbags and seat belts: even with the same change in momentum (stopping in a collision), increasing Δt reduces the average force F = Δp/Δt. This is how impacts are reduced.
Even when force is not constant in an F-t graph (a curved graph), its area still represents impulse. Area calculations are essential in graph problems.
Because momentum is a vector, it is important to choose the sign for direction at the beginning. When a ball hits a wall and rebounds, the change in momentum is 2mv because its direction reverses.
- Momentum p = mv (vector)
- Impulse = FΔt = change in momentum
- Rebound from a wall: sign reversal → change of 2mv
- Area under an F-t graph = impulse
- Impact reduction: increase Δt to distribute the force
- Unit: kg·m/s = N·s
- Derived by integrating F = ma with respect to time
Points to Note (Common Confusions)
① Do not confuse momentum mv with kinetic energy ½mv² (both are quantities related to motion, but they are different). ② Impulse is a vector. Make the sign of its direction clear. ③ The area under an F-t graph is impulse, while the area under a v-t graph is displacement (these are easily confused). ④ Even for a short-duration large force (impact force), knowing FΔt (impulse) is sufficient.
Practice
- A ball of mass 0.20 kg approaches a wall at 15 m/s and rebounds at the same speed. Find the impulse received by the ball.
- In an F-t graph, F = 5.0 N acts from 0 to 4.0 s. Find the impulse and the final velocity of an object of mass 2.0 kg initially at rest.
- Show the relationship between the momentum theorem and the equation of motion F = ma using equations.