Momentum and Impulse
Explore the concepts of momentum and impulse, understand the law of conservation of momentum, and analyse elastic and inelastic collisions.
Momentum
Momentum (p) is the product of an object's mass and velocity. It is a vector quantity -- it has both magnitude and direction. A heavy object moving slowly can have the same momentum as a light object moving fast.
p = mv
Where p = momentum (kg m s-1), m = mass (kg), v = velocity (m s-1)
Tennis Ball
m = 0.06 kg, v = 50 m s-1
p = 3 kg m s-1
Bowling Ball
m = 6 kg, v = 5 m s-1
p = 30 kg m s-1
Car
m = 1500 kg, v = 20 m s-1
p = 30 000 kg m s-1
Important: Momentum is a vector. Two objects with equal mass and speed but moving in opposite directions have momenta that are equal in magnitude but opposite in sign.
Impulse
Impulse (J) is the change in momentum of an object. It equals the force applied multiplied by the time over which it acts. This is derived directly from Newton's second law.
The Impulse-Momentum Theorem
J = FΔt
Impulse = force × time interval
J = Δp = mv - mu
Impulse = change in momentum
Unit of impulse: N s (newton-seconds) or equivalently kg m s-1
Real-world application: Crumple zones in cars increase the collision time, reducing the peak force on passengers. The impulse (change in momentum) is the same, but spreading it over a longer time reduces the force: F = Δp / Δt.
Conservation of Momentum and Collisions
The law of conservation of momentum states that in a closed system (no external forces), the total momentum before an interaction equals the total momentum after. This applies to all collisions and explosions.
m1u1 + m2u2 = m1v1 + m2v2
Total momentum before = Total momentum after
Elastic Collision
Both momentum and kinetic energy are conserved. Objects bounce apart. Example: two billiard balls colliding.
Inelastic Collision
Momentum is conserved, but kinetic energy is not (some is converted to heat, sound, deformation). In a perfectly inelastic collision, objects stick together.
Collisions Summary Diagram
Before Collision
Object A → moving right | Object B stationary
Collision Occurs
Force acts on both objects for time Δt
After Collision
Total momentum unchanged; energy may be lost
Key Vocabulary
Momentum
The product of mass and velocity (p = mv). A vector quantity measured in kg m s-1.
Impulse
The change in momentum of an object, equal to force multiplied by time (J = FΔt). Unit: N s.
Elastic Collision
A collision in which both total momentum and total kinetic energy are conserved.
Closed System
A system in which no external forces act. Total momentum in a closed system is always conserved.
Worked Examples
A 0.5 kg ball moving at 8 m s-1 is hit by a bat and moves at 12 m s-1 in the opposite direction. Find the impulse.
Step 1: Define positive direction as the initial direction. u = +8 m s-1, v = -12 m s-1.
Step 2: J = Δp = m(v - u) = 0.5(-12 - 8) = 0.5 × (-20).
Answer: J = -10 N s (10 N s in the direction opposite to the initial motion).
A 2 kg trolley at 3 m s-1 collides with a stationary 1 kg trolley. They stick together. Find their final velocity.
Step 1: Conservation of momentum: m1u1 + m2u2 = (m1 + m2)v.
Step 2: (2)(3) + (1)(0) = (2 + 1)v ⇒ 6 = 3v.
Answer: v = 2 m s-1 in the original direction of motion.
A 60 kg person jumps from a 200 kg boat at 3 m s-1 relative to the ground. Find the boat's recoil velocity.
Step 1: Initial total momentum = 0 (both at rest). By conservation: 0 = mpersonvperson + mboatvboat.
Step 2: 0 = (60)(3) + (200)vboat ⇒ vboat = -180/200.
Answer: vboat = -0.9 m s-1 (moves opposite to the person).
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
What is the momentum of a 4 kg object moving at 5 m s-1?
Question 2
In which type of collision is kinetic energy conserved?
Question 3
A force of 200 N acts on a ball for 0.05 s. What is the impulse?
Question 4
Why do airbags reduce injury in a car crash?
Question 5
A 5 kg cart at 6 m s-1 collides with a stationary 10 kg cart and they stick together. What is their final velocity?
Key Concepts Summary
- ●Momentum p = mv is a vector measured in kg m s-1.
- ●Impulse J = FΔt = Δp links force, time, and momentum change.
- ●In a closed system, total momentum is always conserved in collisions.
- ●Elastic collisions conserve both momentum and kinetic energy; inelastic conserve only momentum.
- ●Increasing collision time reduces the peak force for the same impulse (safety applications).