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Lecture 11a (Power and Momentum Conservation Examples)

In the absence of external forces on a system the momentum of the system is conserved.

Power is the rate at which energy is delivered, or the rate at which work is done.
The unit of Power is the Watt

definition of power

The output power from an engine is the scalar product of force and velocity

power and speed

The drag force on a car at typical speeds depends on the square of the speed.

definition of drag force

The efficiency of a machine measures the percentage of useful power out to the total power in.

definition of efficiency

Tutorial Questions

Example problems. Try these for practice.

53. A 100 kg person standing on a surface of negligible friction pushes a 50 g stone lying at the person's foot. The stone reaches a forward speed of 3 m.s-1. Find the person's recoil speed.

56. A billiard ball of mass 0.5 kg travelling with speed 7 m.s-1 hits the cushion at 45° and rebounds without loss of speed so that the angle of reflection equals the angle of incidence. Find the (vector) change in the momentum of the ball.

57. A gun on level ground fires a shell giving it a muzzle velocity of 113 m.s-1 at an angle of 60° to the horizontal. When it reaches its maximum height it explodes into two fragments of equal mass. Fragment one, whose speed is immediately zero after the explosion falls directly down. Find:
(a) the time after firing that the shell reaches its maximum height.
(b) the maximum height reached by the shell.
(c) the horizontal distance from the gun that the shell explodes.
(d) the velocity of fragment two immediately after the explosion.
(e) the horizontal distance from the gun that fragment two reaches.

58. A golf ball of mass 50 g is struck a blow that makes an angle of 45° to the horizontal. The ball lands 180 m away on a level fairway. The golf club & ball are in contact for 7 ms. Find
(a) the initial speed of the golf ball
(b) the average force of impact.

59. A bullet of mass 0.1 kg travelling horizontally at 400 m.s-1 is fired into a block of wood at rest on a smooth horizontal table. After the collision the block moves at 20 m.s-1. Find
(a) the initial mass of the block.
(b) the total kinetic energy before and after the collision.

60. A ball of mass 2 kg sliding on a smooth horizontal table with speed 3 m.s-1, makes an elastic collision with a ball of mass 1 kg initially at rest. Find the speeds of both balls immediately after the collision.

E1. A car of mass 1000kg travelling North at 18 is struck by a sports car of mass 400kg travelling West at 108 The cars lock together on collision. Find the resulting velocity (m.s-1) of the combined cars just after the collision.

E2. A 90 kg person is standing still on frictionless ice when hit by a bullet which is stopped in their bullet proof jacket in a time of 5 ms. The accidental shooter fired a pistol with a bullet of mass 8 g that gave the bullet a speed of 394 m s-1. Find
(a) the speed of the person immediately after the hit.
(b) the average force on the person during the 5 ms.

E3. A motorboat runs out of petrol and comes to rest in still water with its front pointed towards the shore 5 m away. The skipper throws a six-pack of beer horizontally from the back at 14 m s-1 relative to the water. The mass of the boat plus crew is 245 kg. The mass of the six-pack is 2.8 kg. Find
(a) the recoil speed of the boat.
(b) the time to reach shore at this speed.

E4. A ute of mass 1732 kg travelling West at 20 m s-1 is struck by a car of mass 800 kg which is travelling South at 25 m s-1. The vehicles lock together on impact. Find the resulting velocity of the combination just after impact.

E5. A 2 kg red ball travelling at 5 ms-1 in the positive x direction collides elastically with a 2 kg green ball travelling also in the positive x direction but moving at 3 ms-1.
(a) state what quantities are conserved in this collision.
(b) write down conservation equations describing the collision.
(c) solve these equations to find the speed of the red ball after the collision.

E6. Two hockey players, each of mass 80 kg and skating at 7 ms-1 collide and lock together. The first player was travelling East and the second player was travelling N390W. Find their combined velocity immediately after the collision.

E7. A 3kg mass is suspended on a massless wire of length 10m. A projectile of mass 2kg travelling horizontally towards the centre of the suspended mass with a speed of 5ms-1 undergoes a straight-line elastic collision with the suspended mass. Find
(a) the velocities of the suspended mass and the projectile immediately after the collision, and
(b) the height to which the suspended mass will rise after the collision.

E8. A gas molecule with mass 4x10-27 kg and speed 30 m.s-1 collides elastically with a second molecule of mass 2x10-27 kg which is initially moving in the same direction at 12 m.s-1. After the collision both molecules move in their initial directions but at different speeds. Find the speeds of both molecules after the collision.

E9. A van of mass 2000 kg is travelling due North at 30 m.s-1 collides with a car of mass 1500 kg travelling 300 South of West at 25 m.s-1. The two vehicles lock together on impact. Find
(a) the velocity of the combined vehicles immediately after the collision, and
(b) the fractional change in the kinetic energy of the vehicles as a result of the collision.

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