Year 11 Physics Momentum and Kinetic Energy

A question asked by students in Year 11 Physics  is "what is the difference between momentum and kinetic energy?" Moving objects have both momentum and kinetic energy but each quantity plays a different role in Physics. A good description of why we need both momentum and kinetic energy is given by Serway and Jewett in Physics for Scientists and Engineers (eighth edition page 236)......There are clear differences between kinetic energy and momentum. First, kinetic energy is a scalar and momentum is a vector. Consider a system of two equal masses heading towards each other on a line with equal speeds. There is kinetic energy associated with this system because members of the system are moving. Because of the vector nature of momentum however, the momentum of this system is zero. A second major difference is  that kinetic energy can transform to other types of energy, such as potential energy or internal energy. There is only one type of linear momentum, so we see no such transformations when using a momentum approach to a problem...

Momentum can be thought of as the tendency of an object to keep moving in a straight line at the same speed. Kinetic energy is  the work done (energy provided) by an external force in moving the object from rest to its final speed.

From a mathematical perspective, in classical physics momentum is defined as mass multiplied by velocity, p=mv and kinetic energy is given by Ek=1/2mv2. Notice that If we differentiate 1/2mv2 with respect to v we get mv which is of course the momentum. This generalisation is one of the first steps in the long road to the development of quantum mechanics. A tutorial sheet on momentum and kinetic energy follows.

  1. Express Ek in terms of p and m.
  2. Two objects P and Q, have masses in the ratio of 2:1 respectively. If each has the same momentum which has the greater kinetic energy?
  3. Two cars are moving along a road. The mass of one car is twice that of the other but it is moving at half the speed of the smaller car. What is the ratio of the kinetic energy of the larger car to that of the smaller car?
  4. Two objects A and B are in motion. The kinetic energy of A is one quarter that of B and the momentum of B is one half that of A. What is the ratio of the speed of A to the speed of B?
  5. A trolley of mass 452g is moving in a straight line on a smooth horizontal laboratory bench. A block of plasticine of mass 146g is initially at rest. Determine the change in kinetic energy of the system when (i) the block is on the bench and the trolley collides with and sticks to the block (ii) the block is dumped on the trolley from a small height as the trolley passes underneath. Describe what has happened in each case to the missing kinetic energy.
  6. A large mass M is at rest on a smooth horizontal table. A smaller mass m moving at a velocity u collides with the larger mass. If the collision is perfectly elastic determine the velocity of the smaller mass after the collision.
  7. In the previous question initially the larger mass M is moving at a velocity U and the smaller mass m is at rest. Determine the velocity of m after the collision.
  8. A trolley of mass M contains a mass m of sand. The loaded trolley moves at a velocity U along a smooth horizontal laboratory bench. The sand starts to leak from the trolley at a constant rate R. Determine the velocity of the trolley when one half of the sand has leaked out.
  9. *Three perfectly elastic spheres of masses m1,m2 and m3 lie in that order and not in contact in a smooth horizontal groove. If m1 is projected towards m2 with velocity U find the velocities of each sphere after two impacts have occurred and show that there will not be a third if m2(m1+m2+m3)>3m1m3