A 75 kg wrestler running east with a speed of 6.0 m s−1crashes into an opponent of mass 100 kg running in the opposite direction at 8.0 m s−1. The two wrestlers collide while in mid-air and remain locked together after their collision.
a Calculate the final velocity of the wrestlers.
b Some time later, the 100 kg wrestler is hurled into the turnbuckle at 5.0 m s−1and comes to a stop. Where has the momentum of this wrestler gone?
Solution
In this problem east will be treated as positive and west will be treated as negative.
a Σpi= Σpf
pi(75 kg wrestler) +pi(100 kg wrestler)=pf(wrestlers locked together) (75 ×6.0) +(100 × −8.0) =(75 +100)vf
450 −800 =175vf
−350 =175vf vf=
= −2.0 m s−1 The final common velocity of the wrestlers is 2.0 m s−1west.
b This momentum (500 kg m s−1) has been transferred to the ring and the Earth, causing the ring and Earth to move slightly.
−350 175
Circus strongmen often perform a feat where they place a large rock on their chest, then allow another person to smash the rock with a sledgehammer.
This might seem at first to be an act of extreme strength and daring. However, a quick analysis using the principle of conservation of momentum will show otherwise. Let us assume that the rock has a mass of 15 kg and that the sledgehammer of mass 3 kg strikes it at 6 m s−1. Using conservation of momentum, we can show that the rock and sledgehammer will move together at just 1 m s−1after the impact.
The large mass of the rock has dictated that the final common speed is too low to hurt the strongman. A more daring feat would be to use the sledgehammer to smash a pebble!
Physics file
• An isolated system is one in which only action/
reaction forces are considered to be acting. External forces (e.g. friction and gravity) are either non-existent or are ignored.
• In any collision in an isolated system, the law of conservation of momentum says that the total
momentum of the system is conserved. The total momentum before the collision is equal to the total momentum after the collision. That is:
Σpi= Σpf
• In a two-body collision, the momentum lost by one body must be gained by the other.
2.2 SUMMARY CONSERVATION OF MOMENTUM
Early in the 20th century an American scientist, Robert Goddard, suggested that a rocket could be sent to the Moon. This view was commonly opposed. People thought that the vacuum in space would give the rocket gases nothing to push against. Goddard set up an experiment in which a gun fired a blank cartridge in a vacuum chamber.
The ‘blank’ produced hot gases and some wadding from the blast. The gun recoiled even though these expelled gases had nothing to push against. Rocket propulsion had been demonstrated.
When a gun fires a bullet, a Newton’s third law action/
reaction pair of forces acts to propel the bullet and to make the gun recoil. The magnitude of the momentum of the gun after the explosion equals that of the bullet. Rockets work on the same principle, except that hot gas replaces the bullet. It seems odd that some hot gas can make a massive rocket move so fast, but you need to keep in mind that the gases are expelled at extremely high speed and that about 90% of the initial mass of the rocket is in fact fuel. The space shuttle is basically a glider attached to several enormous fuel tanks!
The force, or thrust, acting on a rocket can be determined if the exhaust velocity and rate at which the fuel is consumed are known. From the impulse, F∆t=m∆v, so (ignoring the vector nature of the quantities),
F= ∆v
where m/∆t is the rate at which fuel is used and ∆v is the exhaust velocity relative to the rocket.
For example, the first stage of a Saturn V rocket used 15 000 kg of fuel each second, and expelled it at 2500 m s−1 relative to the rocket. The thrust produced by this rocket was:
F= ∆v
=1.5 ×104×2.5 ×103
=3.8 ×107N
As you can see, the forces that the rocket vehicle and the exhaust fuel exert on each other are enormous. This thrust force is opposed by a much smaller force of gravity. If the fuel is expelled at a uniform rate, the thrust acting on the rocket will remain constant. However, this force acts on a decreasing mass as the fuel is consumed. Therefore (as described by Newton’s second law) the acceleration of the rocket will increase as it uses its fuel. During a shuttle launch, the two solid rocket booster tanks are jettisoned after 2 min. These deploy parachutes to descend and are recovered about 250 km away. The large external tank is jettisoned 8 min into the mission and falls into the ocean.
m
∆t m
∆t
Rockets
PHYSICS IN ACTION
Figure 2.13 A space shuttle is attached to giant fuel tanks at launch. As the fuel in these tanks is used, the tanks are discarded. The momentum of the expelled fuel gives the rocket the momentum it needs to accelerate from the ground.
PRACTICAL ACTIVITY 10
Conservation of momentum in explosions
2.2 QUESTIONS
The following information applies to questions 1 and 2. A sports car of mass 1.0 ×103kg travelling east at 144 km h−1 approaches a station wagon of mass 2.0 ×103kg moving west at 72 km h−1.
1 a Calculate the momentum of the sports car.
b Calculate the momentum of the station wagon.
c Determine the total momentum of these vehicles.
2 These two vehicles collide head-on on an icy stretch of highway where there is negligible friction from the road.
The vehicles remain locked together after the collision.
a Calculate their common velocity after the collision.
b Determine the change in momentum of the sports car.
c Determine the change in momentum of the station wagon.
3 A 200 g snooker ball travelling with initial velocity 9.0 m s−1to the right collides with a stationary ball of mass 100 g. If the final velocity of the 200 g ball is 3.0 m s−1to the right, calculate the velocity of the 100 g ball after the collision.
4 A 1000 kg cannon mounted on wheels fires a 10.0 kg shell with a horizontal speed of 500 m s−1. Assuming that friction is negligible, calculate the recoil velocity of the cannon.
5 An arrow of mass 100 g is fired with an initial horizontal velocity of 40 m s−1to the right at an apple of mass 80 g that is initially at rest on a horizontal surface. When the arrow strikes the apple, the two objects stick together.
What is the common velocity of the arrow and apple after the impact?
6 A railway water tanker is rolling freely along train tracks at 5.0 m s−1. A worker opens a plug hole so that water gushes out the bottom of the tanker. Joe and Mary, who were watching, disagreed as to what would happen next.
Joe said that since the tanker was losing mass, its speed would increase in order to comply with the law of conservation of momentum. Mary said that the speed of the cart would not change and momentum would be conserved. Who was correct? Explain.
The following information applies to questions 7–9. A shopping trolley of mass 5.0 kg, carrying 3.0 kg of potatoes and 2.0 kg of oranges, is given an initial push and moves away with constant horizontal speed of 5.0 m s−1. While it is still moving at this speed, another shopper drops a 10 kg bag of apples vertically into the trolley. The effects of friction may be ignored.
7 Determine the subsequent speed of the trolley.
8 Which one of the following statements correctly describes what has happened to the vertical momentum of the apples?
A It has been converted into the horizontal momentum of the trolley.
B It has been dissipated as heat and sound.
C It has been transferred to the Earth.
D It has been converted into kinetic energy.
9 a Which one of the following graphs best represents the horizontal momentum of the trolley alone as a function of time?
b Which one of these graphs best represents the total horizontal momentum of the trolley and its contents as a function of time?
10 A girl of mass 48 kg runs towards her stationary 2.0 kg skateboard and jumps on. The horizontal velocity of the girl just before she makes contact with the skateboard is 4.0 m s−1to the right. Disregard the effects of friction.
a Calculate the momentum of the girl just before she lands on the skateboard.
b Determine the velocity of the girl after she has landed on the skateboard.
c The girl then travels at this speed for a short time before jumping off, still travelling at the same speed as the skateboard. What is the velocity of the skateboard after she jumps off?
11 A footballer crashes into the fence at the MCG and stops completely. What has happened to his momentum? Has momentum been conserved? Explain.
12 As a high-board diver falls toward a diving pool, her momentum increases. Does this mean that momentum is not conserved? Explain.
t
Energy
The Universe is made up of matter and radiation. Matter is the stuff of which things are made—atoms, molecules and subatomic particles. Energy is a property of both matter and radiation. When a force acts on a body, causing it to move, the energy of the body changes. If a water droplet absorbs radiation in the form of microwaves, the water molecules will begin to vibrate strongly and the water will gain heat energy. About 100 years ago, Einstein showed that matter and energy were equivalent and that vast amounts of energy could be obtained from small quantities of matter. You may remember from year 11 how matter is converted into radiation energy during radioactive decay.
Energy comes in many different forms including heat energy, sound energy, light energy, chemical energy, electrical energy and nuclear energy. Energy is a scalar quantity and is measured in joules (J). In year 11, a number of different forms of energy were studied. Some of these are outlined below.
Kinetic energy is the energy of motion. The Earth has a massive amount of kinetic energy as it moves in its orbit around the Sun. An electron has a miniscule quantity of kinetic energy as it orbits the nucleus of an atom.
The KINETIC ENERGY(Ek) of an object is given by:
Ek= mv2 where Ek=kinetic energy in joules (J)
m=mass in kilograms (kg)
v=speed in metres per second (m s−1)
1 2
1000 kg 1000 kg
30 km h–1 60 km h–1
2 kg 0.002 kg
20 m s–1 20 m s–1
(a) (b)
Figure 2.14 (a) The brick and the hailstone are travelling at the same speed, but the brick has 1000 times more kinetic energy because its mass is 1000 times greater. (b) The red car is travelling twice as fast as the green car and has four times as much kinetic energy.
Gravitational potential energy is energy due to an object’s position in a gravitational field. This is a form of stored energy. A cat that is held at rest above the ground has a small amount of gravitational potential energy. As it is held higher from the ground, its gravitational potential energy increases.
GRAVITATIONAL POTENTIAL ENERGY(Ug) is given by:
Ug=mgh
where Ug=gravitational potential energy in joules (J) m=mass in kilograms (kg)
g=gravitational field strength (N kg−1) (g=9.8 N kg−1near the Earth’s surface)
h=height above a reference point in metres (m) 0.2 m
5 kg 2 m
5 kg
Figure 2.15 When the cat is held 2 m off the ground, it has 10 times more gravitational potential energy than when it is only 0.2 m from the ground.
2.3 Work, energy and power
Another form of potential energy that was studied in year 11 is elastic potential energy. This will be covered later in this chapter.