When objects come into contact (or otherwise interact) with each other and exert forces on each other for relatively short periods of time, the interactions can be described as ‘collisions’. We normally expect some or all of the kinetic energy of macroscopic objects to be dissipated in a collision, but it is
important to define the extreme case:
A collision in which the total kinetic energy of the masses is the same before and after the collision is known as an elastic collision.
In our everyday, large-scale world elastic collisions are not possible because some energy is always dissipated into the surroundings.
Momentum is conserved in all collisions and in the theoretical extreme of an elastic collision, kinetic energy is also conserved. The two conservation equations representing this situation can be combined together simultaneously to predict exactly what would happen if an elastic collision occurred (but this is not required in this course). For example, if a mass collides elastically with another identical mass at rest, the only possibility is that the moving mass will stop and the other mass will move off with the same velocity as the first mass. The scientific purpose of a Newton’s cradle (Figure 2.138) is to demonstrate this effect.
Collisions in which some or all kinetic energy is transferred to other forms of energy are called inelastic collisions. All collisions of everyday objects are inelastic.
A collision in which the objects stick together is called a totally inelastic collision.
In an ‘explosion’, masses that were originally at rest with respect to each other are propelled in different directions, so that there is higher kinetic energy after the explosion than before. By definition this type of interaction clearly cannot be described as elastic and can be considered to be similar to a totally inelastic collision in reverse.
The percentage of the total kinetic energy retained in collisions between masses moving together along a straight line can easily be investigated by measuring their masses and their speeds before and after the impact, although a low-friction surface is necessary for reliable results. It is instructive to investigate how the results change when the masses are varied, or the natures of the colliding surfaces are changed. Typically, objects made from elastic materials, like steel and rubber, retain the most kinetic energy. (Remember that a material described as elastic regains its shape after a force has been removed.) Conversely, inelastic materials deform permanently and much energy is dissipated as internal energy and thermal energy.
The concepts of internal energy, sound and deformation cannot be used sensibly to describe individual molecules. Therefore, on the microscopic scale, collisions between particles such as molecules in a gas are usually elastic and easily modelled by computer simulations.
156 A railway truck of mass 8340 kg travelling at 14.3 m s−1 collides with another truck of mass 6420 kg travelling at 8.78 m s−1 in the same direction.
a If after collision the two trucks become joined together, what is their initial speed?
b Calculate the percentage of kinetic energy retained in the collision.
157 A trolley of mass 2.0 kg is moving at a speed of 1.3 m s−1 directly towards another stationary trolley of mass 1.0 kg.
a If immediately after the collision the 1 kg trolley moves at a speed of 1.4 m s−1, what is the speed of the other trolley?
b Calculate the amount of energy dissipated in this collision.
158 If in the previous question the speed of the 1 kg trolley after the collision was stated to be 4.1 m s−1 (instead of 1.4 m s−1), explain why it would still be possible to calculate an answer for part a of the question, but not part b.
159 A cannon of mass 1100 kg fires a cannonball of mass 6.2 kg at a speed of 190 m s−1 (Figure 2.139).
a Calculate the initial recoil speed of the cannon.
b The purpose of firing the cannon is to transfer chemical energy of the explosive into kinetic energy of the cannon ball, but the cannon is also given kinetic energy. Calculate the percentage of the total kinetic energy that is carried by the cannon ball.
Examination questions – a selection
Paper 1 IB questions and IB style questions
Q1 M is a small mass on the end of a lightweight string. It has been pulled to one side with a force that keeps it stationary.
Which of the following four diagrams is the correct free-body representation of the forces acting on the mass?
Q2 An object of weight W is slipping down a slope (inclined plane) that makes an angle of θ with the horizontal. If the object is moving with constant speed, what is the magnitude of the frictional force up the slope?
A W B W sin θ C W cos θ D
Q3 The work done when a constant force acts on a mass is always equal to:
A the magnitude of the force multiplied by the distance moved by the mass
B the magnitude of the force multiplied by the displacement perpendicular to the force
C the magnitude of the force multiplied by the displacement in the direction of the force D the vector sum of the force and the distance moved by the mass.
Q4 A rocket is travelling across space when its engine ejects gases of total mass m in a time t. The speed of the gases relative to the rocket is v.
Which of the following is the correct expression for the force exerted by the gases on the rocket?
A mv B C D
Q5 Which of the following is a correct definition of the instantaneous velocity of a moving object at time t?
A B
C rate of change of displacement at time t D rate of change of distance at time t
Q6 A big object is dropped from a large height. It hits the ground at time T after being dropped. Which of the following graphs best represents how the speed, v, of the object varies with time, t, until just before it hits the ground?
A
B
C
D
Q7 When the motion of two cars was compared, it was found that car A was more powerful than car B.
Which of the following statements must be true?
A Car A produces more useful energy than car B.
B Car A produces a greater force than car B.
C In the same time, car A does more useful work than car B.
D In the same time, car A moves a greater distance than car B.
Q8 A mass of 5 kg is pulled up a slope at a constant speed of 2 m s−1. After rising a vertical height of 4 m, the total work done was 1200 J. The work done in overcoming friction was:
A 1000 J B 200 J C 2400 J D 1400 J
Q9 An increasing force acts on an object and its acceleration increases as shown in the graph.
If the object was initially at rest, what is the speed of the object after 20 seconds?
A 0.5 m s−1
B 2.0 m s−1 C 100 m s−1 D 200 m s−1
Q10 If there is no resultant force acting on an object, which of the following quantities must also be zero?
A speed B velocity C acceleration D momentum
Q11 An electric motor raises a 2.5 kg mass a distance of 12 m in a time of 6 s. If the efficiency of the process was 20%, what was the input power to the motor?
A 10 W B 25 W C 250 W D 500 W
Q12 A mass is moving at a constant speed with a kinetic energy EK. What is the kinetic energy of another mass that has twice the mass and half the speed of the first mass?
A B EK C 2EK D 4EK
Q13 A vehicle is driven up a hill at constant speed. Which of the following best describes the energy changes involved?
A Chemical energy is converted into gravitational potential energy.
B Chemical energy is converted into gravitational potential energy, sound and thermal energy.
C Gravitational potential energy is converted into chemical energy.
D Gravitational potential energy is converted into chemical energy, sound and thermal energy.
© IB Organization Q14 A weight W is suspended from the ceiling on the end of a length of string. According to Newton’s
third law of motion there must be another force equal and opposite to the weight. This second force is:
A the downwards force of the string on the ceiling B the upwards force of the string on the weight
C the upwards force exerted by the weight on the Earth D the tension in the string.
Q15 A stone is thrown up into the air at an angle to the horizontal. Assuming that air resistance is negligible, which of the following is not constant while the stone is moving through the air?
A horizontal component of velocity B vertical component of velocity C total energy of the stone
D acceleration of the stone
Q16 A steel ball is released from rest into a cylinder containing oil. Which of the following statements is incorrect?
A The force opposing motion is called drag.
B If the cylinder is large enough the ball will reach a terminal speed.
C The weight of the ball is reduced in the oil.
D A larger ball will experience a greater resistive force.
Q17 When an unstretched steel spring was extended by 10 cm the elastic potential energy stored in it was 0.20 J. What was the force constant of the spring?
A 4.0 × 10−3 N m−1 B 4.0 N m−1
C 10 N m−1 D 40 N m−1
Q18 When a ball was dropped onto a hard surface a student believed that the collision was elastic. For this to be true, the ball must:
A bounce up to the same height from which it was dropped B stretch a lot
C be made of rubber D get hotter.
Q19 A gas atom strikes a wall with speed v at an angle θ to the normal to the wall. The atom rebounds at the same speed v and angle θ.
Which of the following gives the magnitude of the momentum change of the gas atom?
A zero B 2mv sin θ C 2mv D 2mv cos θ
© IB Organization Paper 2 IB questions and IB style questions
Q1 A bullet of mass 32 g is fired from a gun. The graph shows the variation of the force F on the bullet with time t as it travels along the barrel of the gun.
The bullet is fired at time t = 0 and the length of the barrel is 0.70 m.
a State and explain why it is inappropriate to use the equation s = ut + ½at2 to calculate the acceleration of the bullet.
(2) b Use the graph to:
i determine the average acceleration of the bullet during the final 2.0 ms of the graph
(2) ii show that the change in momentum of the bullet, as the bullet travels along the length of the
barrel, is approximately 9 N s.
(3) c Use the answer in b ii to calculate:
i the speed of the bullet as it leaves the barrel
(2) ii the average power delivered to the bullet.
(3) d Use Newton’s third law to explain why a gun will recoil when a bullet is fired.
(3)
© IB Organization Q2 A clay block initially on the edge of a table is fired away from the table, as shown in the diagram.
The initial speed of the clay block is 4.3 m s−1 horizontally. The table surface is 0.85 m above the ground.
a Ignoring air resistance, calculate the horizontal distance travelled by the clay block before it strikes the ground.
(4) b The diagram shows the path of the clay block neglecting air resistance. Make a copy of the
diagram and show on it the approximate shape of the path that the clay block will take assuming that air resistance acts on the clay block.
(3)
© IB Organization Q3 a A system consists of a bicycle and cyclist travelling at a constant velocity along a horizontal road.
i State the value of the net force acting on the cyclist.
(1) ii On a copy of the diagram, draw labelled arrows to represent the vertical forces acting on the
bicycle.
(2) iii With reference to the horizontal forces acting on the system, explain why the system is
travelling at constant velocity.
(2) b The total resistive force acting on the system is 40 N and its speed is 8.0 m s−1. Calculate the
useful power output of the cyclist.
(1)
c The cyclist stops pedalling and the system comes to a rest. The total mass of the system is 70 kg.
i Calculate the magnitude of the initial acceleration of the system.
(2) ii Estimate the distance taken by the system to come to rest from the time the cyclist stops
pedalling.
(2) iii State and explain one reason why your answer to c ii is only an estimate.
(2)
© IB Organization Q4 a Explain the difference between the coefficients of static and dynamic coefficients of friction.
(2) b The angle of a flat wooden slope to the horizontal is increased slowly until a metal cube just
begins to slide down the surface. If the coefficient of static friction between the surfaces is 0.56, what is the greatest possible angle before the cube begins to slide?
(2)