Hon Lam 翰林書齋 F.4 Physics

MOMENTUM

If a vehicle collides with a wall, which causes greater damage ?

(a) A vehicle of a ------------- velocity causes greater damage.

(b) A vehicle of a ------------- mass causes greater damage.

The above arguments suggest that the amount of motion of an object depends both its mass and velocity. The quantity of motion is called linear momentum (or momentum) which is defined as :

Momentum = mass x velocity

P = m v

SI unit : kg m s

Direction : in the same direction as velocity

Example

List the following moving objects in the order of increasing momentum :

(a) a car travelling at 70 km h ,

(b) a man running at 30 km h ,

(d) a golf ball travelling at 100 km h .

CONSERVATION OF MOMENTUM

A. Collisions

There are three types of collisions.

(1) Inelastic collision (Completely inelastic collision)

Two bodies collide and then stick together. Part of the initial mechanical energy of the colliding bodies will convert into internal energy and sound energy, or even energy for deforming the bodies.

e.g. The collision between two hanging plasticine balls

(2) Partially elastic collision

Under most circumstances, two bodies collide and separate after the impact. Part of the initial mechanical energy of the colliding bodies will convert into internal energy and sound energy, or even energy for deforming the bodies.

e.g. The collision between two hanging wooden balls

(3) Elastic collision (Completely elastic collision)

Two bodies collide each other. They will resume their shapes and bounce away with their relative speeds unchanged. There is no change in mechanical energy.

e.g. The collision between two hanging silicone balls

Strictly speaking, a perfectly elastic collision does not exist in the macroscopic world. Nevertheless, it is the most common type of collision that exists between atoms.

B. Conservation of momentum

A physical law governs the velocities of colliding objects before and after a collision. We will investigate this by experiments using the ticker-tape timer and trolleys.

(1) Inelastic collisions

Compensate the runway for friction. Fix two pins on trolley A and two corks on trolley B so that they stick together on collision. Give trolley A a sharp push to move it down the runway and collide with trolley B.

Repeat with two stacked trolleys and then three stacked trolleys colliding with a stationary trolley.

Conclusion : In an inelastic collision, the momentum before collision is equal to the momentum after collision. That is, momentum is conserved.

(2) Elastic collision

Compensate the runway for friction. Release the plunger in trolley A so that when it collides with trolley B, the two trolleys move off separately. The tapes attached to the two trolleys pass through the same timer but each has a carbon disc. Push one, two and three stacked trolleys in turn to collide with a stationary trolley.

Results :

Conclusion : After allowing for errors, the total momentum before collision is the same as that after collision. Momentum is also conserved in elastic collisions.

(3) “Explosion” of trolleys

Place two trolleys in contact end-on and release the plunger suddenly. The two trolleys “explode” apart, moving off in opposite directions. Repeat using two and then three stacked trolleys placed end-on with one trolley.

Results :

Before explosion, total momentum =

After explosion :

Conclusion : Momentum is also conserved in an “explosion”.

(4) Momentum is conserved in all the three experiments above. This is the principle of conservation of momentum which states that :

In a collision the total momentum of the colliding objects before collision is equal to the total momentum after collision, provided that there is no external force acting on the objects.

m u + m u = m v + m v

Example

A motorcycle of mass 200 kg is moving at 30 m s on a highway. A truck of mass 4000 kg is moving at 10 m s ahead. An accident occurs when the motorcycle hits the truck. Immediately after the collision, the truck moves forward at 11.8 m s .

(a) Find the velocity of the motorcycle immediately after the collision.

(b) What is the change in momentum of the motorcycle ?

Example : Recoil velocity

When a bullet is fired from a rifle, the rifle recoils. There is a forward momentum of the bullet. This has to be balanced by a backward momentum of the rifle, so that momentum is conserved.

Now a bullet of mass 12 g is fired from a rifle of mass 5 kg with a speed of 150 m s . Find the recoil velocity of the rifle.

Example : Disappearing momentum

(a) A piece of cardboard placed on a thin layer of polystyrene beads on a bench top. When a “push-and-go” toy car runs forward on the cardboard, what happens to the cardboard ?

The cardboard moves -----------------------.

Since the initial momentum of the car and the cardboard is zero, by the principle of conservation of momentum, the final momentum is also zero. Thus the forward momentum of the toy car is equal and opposite to the backward momentum of the cardboard.

(b) If a girl step from a rowing boat onto the bank, what happens to the boat ?

The boat moves ---------------.

Since the initial momentum of the girl and the boat is zero, by the principle of conservation of momentum, the final momentum is also zero. Thus the forward momentum of the girl is equal to and opposite to the backward momentum of the boat.

The road should move backward slightly. But since the mass of the earth is very great, such motion cannot be detected. Thus the momentum seems to have disappeared.

Example : Colliding balls

The principle of conservation can be demonstrated by using a toy called “Newton’s cradle”. Pull one ball to one side and find out what happens when it collides with the other four balls. Repeat using two, three and then four balls.

Reasons : This is accordance with the law of conservation of momentum : the total momentum of balls before collision is equal to the total momentum of balls after collision.

Example

A bullet of mass 4 g is fired into a lump of plasticine on a stationary trolley of mass 0.996 kg. The bullet is embedded in the plasticine and the trolley takes 1.25 s to move along a 1 m friction-compensated track.

Calculate the speed of the bullet.

Example

A girl of mass 40 kg standing still on a pair of ice skates throws a ball of mass 1 kg horizontally at a velocity of 4 m s . At what velocity does the girl move ?

Example

There are two trolleys A and B standing at rest on a friction-compensated runway. A is given a push and collides with B. The ticker-tapes obtained from the experiment are as shown. The ticker-tape timers produce 50 dots per second.

(a) Using the scale on the metre as shown, choose a suitable portion of the tapes and find the speed of

(1) A before the collision,

(2) A after the collision, and

(3) B after the collision.

(b) Find the ratio of the mass of A to that of B from (a).

If you were the teacher, how would you show that the result are impossible ?

CHANGE IN MOMENTUM (IMPULSE)

Newton’s second law can be expressed in terms of the change in momentum.

A. Change in momentum

An unbalanced force F acts on an object of mass m changing its velocity from u to v in time t.

From Newton’s second law,

F = ma

Hence Newton’s second law can be restated as :

The unbalanced force acting on an object is equal to the rate of change of momentum of the object.

Change in momentum

Force = ----------------------------

Time taken

Alternately,

F t = m v - mu

Impulse = change in momentum

The product F t is called the impulse acting on the object.

S.I. unit of impulse : N s or kg s

B. Time of impact

(1) Hitting a ball

When a tennis ball is hit, a large force acts on it for a very short time interval which may be a few millisecond (ms). If you know the time of impact and the velocity of the ball leaving the racket, you can work out the force on the ball.

The time of impact depends on the hardness of the object. The harder it is the shorter is the time of impact.

For example, a golf ball is much harder than a tennis ball – it has a shorter time of impact on being struck by a golf club.

Example

A 0.2 kg tennis ball travelling at 10 m s is hit by a racket and returns at 20 m s . The ball is in contact with the racket for a time of 0.1 s. Find

(a) the change in momentum of the ball;

(b) the average force exerted on the ball by the racket.

(2) Jumping

If we increase the time of impact, the average force is then reduced.

(a) When we jump from a height to the ground, we bend our knees slightly. This action will bring our bodies to rest in a longer time and so reduce the force of impact which otherwise hurt the joints of our bodies.

(b) In high jump and pole vault, soft mats are placed on the ground to increase the time of impact. This action will bring our bodies to rest in a longer time and so reduce the force of impact.

(3) Car safety

If we increase the time of impact, the average force is then reduced.

(a) A car is fitted with bumpers. The passenger cabinet is strong but two ends are collapsible. During a car crash, the bumpers and the ends collapse. The impact time is increased and the possibility of injury to the passengers is much reduced.

(b) The seat-belt is another safety device. When the car stops suddenly, the passenger is thrown forward but is stopped by the seat-belt. The seat belt is designed to stretch if a collision happens. This increases the stopping time of the passenger and so reduces the force of impact.

Example

A car of mass 1000 kg moving at 72 km h collides with a wall head-on and comes to a stop. Calculate the force of impact on the car

(a) if the car has a rubber bumper and a collapsible front section, and it stops in 0.4 s.

(b) If the car has a strong bumper and it stops in 0.02 s.

Example

A van travelling at 20 m s collides with a wall head-on and is brought to stop. Whatis the force on the driver of mass 80 kg.

(a) if he wears a seat belt and comes to a stop in 1.0 s.

(b) if he does not wear a seat belt and is thrown forward coming to stop in 0.05 s.

(4) Falling objects can kill

A free falling object accelerates at 10 m s . It can reach a very high speed even after falling a short distance. If it lands on someone it may cause serious or fatal injuries.

Example

A can of soft drink is dropped from the tenth floor of a building as shown. Taking g as 10 m s .

Calculate

(a) the speed of the can just before it strikes the floor, and

(b) the average force the can exerts on the floor, assuming that it was stopped by the floor in 0.02 s and that it did not rebound.

NEWTON’S THIRD LAW

Newton’s third law states that

If object A exerts a force on object B, then object B will exert an equal but opposite force on A.

A. Action-and-reaction pairs

(1) If a roller-skater pushes against the wall, he moves away from it. The roller-skater exerts a force on the wall; the wall exerts a force on the roller skater.

(2) A boy and a girl each sits on a trolley holding one end of a rope. If the boy alone pulls, they move towards each other. If girl alone pulls, they again move towards each other. Since the boy and the girl exert an equal and opposite force on each other.

(3) Two trolleys “explode” apart when the spring-loaded plunger of one trolley is released. The two trolleys move off in opposite directions with the same velocity. Since the two trolleys exert an equal and opposite force on each other.

(4) A boy is standing on the floor.

(a) The earth pulls the boy downwards and the floor pushes him upwards.

The two forces are equal and opposite but they do NOT belong to the same action-and-reaction pair.

Since the two trolleys exert an equal and opposite force on each other.

(b) The boy exerts a downward force on the floor and the floor exerts an upward force on the boy – these two forces form one action-and-reaction pair.

(5) Rockets

Partially fill the water rocket with water and pump air into it.

Pull the trigger and the compressed air will force the water out and lift the rocket off.

The compressed air exerts an equal and opposite force on the air and shoots the rocket up.

The rocket engine works on the same principle, but, instead of water, it pushes out large masses of hot gas.

The gas is produced by mixing the fuel (liquid hydrogen) with liquid oxygen in the combustion chamber and burning the mixture fiercely.

(6) Jets

A jet engine does not carry a supply of liquid oxygen. Instead, it uses a compressor to draw in large masses of air from the atmosphere. The fuel is mixed with the compressed air and is burnt to produce large masses of hot gas which are ejected.

Example

A rocket, taking off vertically pushes out 20 kg of gas every second at a velocity of 150 m s .

(a) What is the upward thrust on the rocket ?

(b) If the total mass of the rocket is 250 kg, what is the downward gravitational force ?

(d) What is the upward acceleration ?

(e) What is the acceleration of the rocket when it has burned off 100 kg of fuel ?

B. Momentum and Newton’s laws

By the principle of conservation of momentum,

By Newton’s second law,

This is Newton’s third law.

Example

A ball of mass 0.1 kg is dropped from a height of 20 m above the ground.

It rebounds to a height of 12 m. On impact the ball is in contact with the ground for 0.02 s. Assuming that air resistance is negligible, find

(a) the speed of the ball on hitting the ground,

(b) the speed of the ball on leaving the ground,

(2) as it leaves the ground,

(d) the change in momentum of the ball,

(e) the average force acting on the ball during impact.

Example

A man stands in a hanging crate as shown. The man weighs 800 N and the crate has a weight of 200 N. When the man pulls on the rope, the force he exerts in the crate is 550 N.

(a) Draw a free body diagram of the man and the crate.

(b) Find the acceleration of the crate.

Find the tension in the rope.

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