Conservation of Momentum

Σpbefore = Σpafter Conservation of Momentum In a collision or an explosion where there is no external force, the total momentum is conserved. “The total momentum before equals the total momentum after” Σpbefore = Σpafter

“Before” Collision A B (Collision) A B “After” Collision A B

Σpbefore (Total momentum “before”) “Before” Collision A B Σpbefore (Total momentum “before”) Σpbefore = Momentum of ball A “before” + Momentum of ball B “before” Σpbefore = pAi + pBi = mAvAi + mBvBi

Σpafter (Total momentum “after”) “After” Collision A B Σpafter (Total momentum “after”) Σpafter = Momentum of ball A “after” + Momentum of ball B “after” Σpafter = pAf + pBf = mAvAf + mBvBf

*Sometimes this is written as: Conservation of Momentum Principle says: “Total momentum before = Total momentum after” Σpbefore = Σpafter which means: pAi + pBi = pAf + pBf mAvAi + mBvBi = mAvAf + mBvBf *Sometimes this is written as: mAuA + mBuB = mAvA + mBvB

pA(before) + pB(before) = pA(after) + pB(after) Total momentum before = Total momentum after pA(before) + pB(before) = pA(after) + pB(after) mAuA + mBuB = mAvA + mBvB Before After

A spectacular collision Two elastic kids, Aaron (200 kg) and Brad (300 kg) are running towards each other. Aaron is running at 10 ms-1 and Brad is running at 8.0 ms-1. Moments later, they collide and Aaron bounces back at 8.9 ms-1 and Brad bounces back at 4.6 ms-1. Calculate the total momentum (of Aaron and Brad) before the collision. Calculate the total momentum after the collision.

Example: Two balls, A and B, are rolling towards each other and are about to collide. Ball A has a mass of 400 g and is travelling at 0.8 ms-1. Ball B has a mass of 300 g and is travelling at 0.7 ms-1. Calculate the total momentum “Before”

The two balls have collided and now are moving away from each other. What is the total momentum now? If the momentum of ball A immediately after the collision is 0.1944 kgms-1, calculate: the momentum of ball B the velocity of ball B “After”

When two objects become connected A shunting locomotive (m = 5000 kg) travelling at 3 ms-1 bumps into and locks onto a stationary carriage of mass 3000 kg. What combined speed do the locomotive and carriage have after the collision?

When two objects become connected A Morris Minor car (mass 750 kg) is travelling at 30 ms-1 and collides head-on with a Mercedes Benz car (mass 1600 kg) travelling at 20 ms-1 in the opposite direction. The two cars lock together in the crash. Calculate the total momentum Calculate the velocity of the combined wreckage after the collision Would the wreckage keep moving at this velocity? Why or why not?

In an Explosion (instead of a collision) A rifle (m 4 kg) fires a bullet (m = 20 g) which travels at a speed of 400 ms-1. What is the recoil speed of the rifle?

Worksheets 10 & 11 Activity 10B Questions 1 ~ 6

A spectacular collision revisited Two elastic kids, Aaron (200 kg) and Brad (300 kg) are running towards each other. Aaron is running at 10 ms-1 and Brad is running at 8.0 ms-1. Moments later, they collide and Aaron bounces back at 8.9 ms-1 and Brad bounces back at 4.6 ms-1. Calculate the change in momentum of Aaron. Calculate the change in momentum of Brad.

Conservation of Momentum and Impulse (Δp) Two balls, A and B, are rolling towards each other. Ball A (m = 500 g) is moving at 1.0 ms-1 and ball B (m = 1.5 kg) is moving at 0.50 ms-1. After they collide, Ball A moves in the opposite direction at 1.25 ms-1. Calculate: the change in momentum of ball A (ΔpA) the change in momentum of ball B (ΔpB)

Conservation of Momentum and Impulse (Δp) In a collision between two objects, the change in momentum (impulse) of one object is equal to the change in momentum of the other object, only in the opposite direction. (ΔpA = -ΔpB) This is because when the two objects collide, they apply the same force on each other, in the opposite direction. (Newton’s 3rd Law!! )

Example A boy racer is travelling at 25 ms-1. He crashes into the back of another car which is travelling at a speed of 12.5 ms-1 in the same direction. Boy racer’s car = 1200 kg The other car = 2000 kg Find the total momentum. The speed of the boy racer’s car immediately after the crash is 9.37 ms-1. Calculate the speed of the other car immediately after the crash. Calculate the change in momentum of each car.

A spectacular collision revisited again Two elastic kids, Aaron (200 kg) and Brad (300 kg) are running towards each other. Aaron is running at 10 ms-1 and Brad is running at 8.0 ms-1. Moments later, they collide and Aaron bounces back at 8.9 ms-1 and Brad bounces back at 4.6 ms-1. Calculate the total kinetic energy (of Aaron and Brad) before the collision. Calculate the total kinetic energy after the collision.

Think about this: Conservation of momentum principle says: “In a collision or an explosion where there is no external force, the total momentum is always conserved” How about kinetic energy? Would total kinetic energy be conserved in a collision as well? (ΣKE(before) = ΣKE(after) ?)

Collisions – elastic vs. inelastic One may ask – “Couldn’t we just use the conservation of energy principle, by calculating the total kinetic energy?” The answer is NO – because often some energy is lost due to friction, meaning that the “total kinetic energy” is not conserved. However, there are collisions where “total kinetic energy” IS conserved. These collisions are called “elastic collisions”.

Total Ek(before) = Total Ek(after) An elastic collision is a collision which: the total kinetic energy before equals the total kinetic energy after. (The total kinetic energy is conserved.) Total Ek(before) = Total Ek(after) *If the total kinetic energy is NOT conserved, the collision is inelastic. In fact, most collisions are inelastic.

Example Two objects of equal mass, A and B, are travelling towards each other. Object A (2 kg) is moving at 0.6 ms-1 and object B is moving at 0.5 ms-1. After they collide, the two objects travel in the opposite direction. Object A travels at 0.39 ms-1 and object B travels at 0.49 ms-1. Is the total momentum conserved? Is the total kinetic energy conserved? Is this collision elastic or inelastic?

Worksheets 10 & 11 Activity 10B Questions 1 ~ 6 Homework Booklet: “Momentum”