3aLinear Momentum
Define momentum as a vector (p = mv), apply conservation of momentum to isolated systems, and analyze collisions and explosions.
Momentum conservation is one of the most powerful principles in physics — it explains everything from car crashes to rocket propulsion to subatomic particle collisions. It is a direct consequence of Newton's third law.
Why is it harder to stop a moving truck than a moving bicycle at the same speed — and why does a cannon recoil when it fires a cannonball?
Lesson Overview
Linear momentum (p = mv) is a vector quantity that measures the "quantity of motion" of an object. It depends on both mass and velocity. The law of conservation of momentum states that the total momentum of an isolated system (no net external force) remains constant. This powerful principle applies to collisions, explosions, and any interaction between objects. Because momentum is a vector, direction matters — momentum in one direction can cancel momentum in the opposite direction.
Key Equations
Worked Examples
Find the momentum of (a) a 1500 kg car moving at 20 m/s east, and (b) a 0.15 kg baseball moving at 40 m/s west.
A 2 kg ball moving at 5 m/s east collides with a 3 kg ball at rest. After the collision, the 2 kg ball moves at 1 m/s east. Find the velocity of the 3 kg ball after the collision.
A 5 kg cannon fires a 0.5 kg cannonball at 100 m/s. Both are initially at rest. Find the recoil velocity of the cannon.
A 70 kg skater moving at 3 m/s east catches a 2 kg ball thrown at 10 m/s west. Find the skater's velocity after catching the ball.
A 1000 kg car moving at 15 m/s north and a 1500 kg truck moving at 10 m/s north collide and stick together. Find their combined velocity.
Guided Problems
Find the momentum of a 0.5 kg hockey puck moving at 30 m/s. Then find the momentum of a 2000 kg truck moving at 0.5 m/s. Which has greater momentum?
Hint: Use p = mv for each. Compare the results — a slow massive object can have more momentum than a fast light one.
A 3 kg object moving at 4 m/s east collides with a 5 kg object moving at 2 m/s west. They stick together. Find their combined velocity after the collision.
Hint: Assign east as positive. p_before = 3(4) + 5(−2). Set equal to (3+5)v_after and solve.
A 60 kg person standing on a frictionless ice rink throws a 2 kg ball at 8 m/s east. Find the person's recoil velocity.
Hint: Initial momentum is zero. Use conservation: 0 = m_ball × v_ball + m_person × v_person. Solve for v_person.
A 1200 kg car moving at 25 m/s has its brakes applied, bringing it to rest. Find the change in momentum of the car.
Hint: Δp = p_f − p_i = 0 − mv_i. The change in momentum is negative (opposite to initial motion).
Two ice skaters (m₁ = 60 kg, m₂ = 80 kg) push off from each other from rest. Skater 1 moves at 4 m/s east. Find skater 2's velocity.
Hint: Initial momentum = 0. After: m₁v₁ + m₂v₂ = 0. Solve for v₂.
Key Vocabulary
Linear Momentum (p)
The product of an object's mass and velocity: p = mv. It is a vector quantity with the same direction as velocity. Units: kg·m/s.
Example: A 1500 kg car at 20 m/s has p = 30,000 kg·m/s — much more than a 0.15 kg baseball at 40 m/s (p = 6 kg·m/s).
Conservation of Momentum
The total momentum of an isolated system (no net external force) remains constant. For two objects: m₁v₁ᵢ + m₂v₂ᵢ = m₁v₁f + m₂v₂f.
Example: When a cannon fires, the cannonball gains forward momentum and the cannon gains equal backward momentum — total stays zero.
Isolated System
A system on which no net external force acts. Internal forces (between objects in the system) do not change the total momentum.
Example: Two colliding billiard balls form an isolated system if friction is negligible — their total momentum is conserved.
Newton's Second Law (momentum form)
The net force equals the rate of change of momentum: F_net = Δp/Δt. This is actually Newton's original formulation.
Example: A 1000 N force acting for 2 s changes an object's momentum by Δp = FΔt = 2000 kg·m/s.
Workbook Check — Interactive Quiz
Interactive Practice — 5 Questions
A 2 kg ball moves at 5 m/s east. What is its momentum?
A 4 kg object moving at 3 m/s east collides with a 2 kg object at rest. They stick together. What is their combined velocity?
A 5 kg cannon fires a 0.1 kg bullet at 300 m/s. Both start at rest. What is the cannon's recoil speed?
Which of the following has the greatest momentum?
The total momentum of an isolated system is conserved. What does "isolated" mean?
Independent Practice
Find the momentum of a 0.145 kg baseball moving at 44 m/s (a typical fastball).
A 1200 kg car moving at 20 m/s collides with a stationary 800 kg car. They stick together. Find their combined velocity.
Two ice skaters (60 kg and 80 kg) push off from rest. The 60 kg skater moves at 3 m/s east. Find the 80 kg skater's velocity.
A 0.5 kg ball moving at 6 m/s east collides with a 1.5 kg ball moving at 2 m/s west. They stick together. Find the final velocity.
★ A 70 kg astronaut in space throws a 2 kg wrench at 5 m/s to the right. (a) Find the astronaut's recoil velocity. (b) If the astronaut then throws a second 2 kg wrench at 5 m/s to the right, find the astronaut's new velocity. (c) What is the astronaut's total momentum after both throws?
ChallengeCommon Mistakes
Forgetting that momentum is a vector — ignoring direction
Always assign positive/negative directions. Momentum to the left is negative if right is positive. Add momenta with signs
Applying conservation of momentum when external forces act
Conservation of momentum only applies to isolated systems. If friction, gravity, or normal forces have a net external component, momentum is not conserved
Confusing momentum (p = mv) with kinetic energy (KE = ½mv²)
Momentum is a vector proportional to v; kinetic energy is a scalar proportional to v². They are different quantities with different units
Thinking the heavier object always has more momentum
Momentum depends on both mass AND velocity. A fast light object can have more momentum than a slow heavy one
Math Tips
p = mv is a vector. Always define a positive direction and assign signs to velocities before calculating.
Conservation of momentum: Σp_before = Σp_after. Write out each term: m₁v₁ᵢ + m₂v₂ᵢ = m₁v₁f + m₂v₂f.
For explosions (cannon, rocket, push-off): initial momentum = 0, so final momenta must sum to zero: m₁v₁ + m₂v₂ = 0.
Newton's 2nd law in momentum form: F_net = Δp/Δt. This is more general than F = ma and works even when mass changes.