Concepts: collisions and exercise practice
Work, Energy and Power24 min readLesson 4 of 6
4.1 Collisions in one dimension
In a collision between two bodies with no external force, momentum is conserved. The kinetic energy may or may not be conserved, and the two cases are called elastic and inelastic. In an elastic collision the relative speed of approach equals the relative speed of separation, so the coefficient of restitution is one. In a perfectly inelastic collision the bodies stick together and move with a common velocity, and the loss of kinetic energy is maximum while momentum is still conserved.
The standard results are worth remembering. For an elastic head on collision between masses m1 and m2, the final velocity of the first is (m1 minus m2) over the sum, times u1, plus twice m2 over the sum, times u2. When the masses are equal the bodies simply exchange velocities, which is seen on a billiard table and in Newton's cradle. When a light body strikes a heavy one at rest it rebounds with nearly its own speed reversed.
The perfectly inelastic result is simpler: both bodies move with the total momentum divided by the total mass. A bullet embedding in a wooden block and two lumps of clay sticking together are examples. The kinetic energy lost in sticking is converted into deformation and heat, and it equals the initial kinetic energy minus the kinetic energy of the combined body.
4.2 Exercise practice: work and energy
Exercise 5.1 asks for the work done by a force given as a function of position, which is the integral of the force. Exercise 5.2 computes the work done by gravity and by friction for a block sliding down an incline and checks it against the change in kinetic energy. Exercise 5.3 uses the work-energy theorem to find a speed after a known work, and exercise 5.4 uses the area under a force-displacement graph.
Exercises 5.5 and 5.6 ask for the change in kinetic and potential energy in a lift and in a freely falling body, and exercise 5.7 checks the conservation of mechanical energy for a ball thrown upward. Exercise 5.8 gives a potential energy function and asks for the force as its negative slope. Exercises 5.9 and 5.10 use the spring energy to find a compression and a launch speed, and exercise 5.11 finds the work done in stretching a spring from one length to another.
Exercises 5.12 and 5.13 are power numericals, one for a pump lifting water and one for a machine lifting a load, with efficiency in the second. Exercise 5.14 compares the power needed to walk up a slope with the power needed to run up it, and exercise 5.15 compares the kinetic energies of two bodies with equal momentum. Exercises 5.16 and 5.17 use the potential energy curve to find turning points and equilibrium positions.
4.3 Exercise practice: collisions and power
Exercise 5.18 is a one dimensional elastic collision between two bodies, solved with momentum and energy or with the coefficient of restitution. Exercise 5.19 is a perfectly inelastic collision in which a bullet embeds in a block, and exercise 5.20 asks for the loss of kinetic energy when two equal masses collide and stick. Exercise 5.21 uses a ballistic pendulum, where the bullet's momentum is conserved in the embedding and the energy is conserved in the swing, in two stages.
Exercise 5.22 asks for the height reached by a body thrown upward from a moving vehicle, an exercise in the reference frame and in energy conservation. Exercise 5.23 combines power with the work done against gravity for a climber. Through all of these, the choice of tool is what matters: momentum for the collision, energy for the motion after it, and power for the rate.
A useful check on a collision answer is that the final kinetic energy can never exceed the initial kinetic energy, and for a perfectly inelastic collision it is always smaller. In an elastic collision of equal masses the kinetic energy is transferred completely. Answers that invent energy are wrong even when the algebra looks tidy.
Exam
In a ballistic pendulum, do not set kinetic energy equal before and after the embedding. Momentum is conserved during the embedding and energy is not; energy is conserved only during the swing afterwards.
4.4 Choosing between momentum and energy
Momentum conservation applies to any system with no external force, whether the collision is elastic or not. Energy conservation applies to the motion after the collision, where the forces are conservative. Mixing the two stages is the standard trap: the embedding is a momentum event with energy loss, and the swing is an energy event with no momentum conservation because gravity acts.
For elastic collisions the two conservation laws together give the final velocities. For inelastic collisions only momentum survives and the energy loss must be computed. For explosions the same momentum logic runs backwards: a body at rest splits into parts whose momenta cancel, and the kinetic energy comes from stored energy rather than from an external source.
Collision problems often carry a figure, and the direction of each velocity must be taken from it before the signs are written. A head on collision has all velocities along one line; a two dimensional collision needs both components of momentum. In every case, write the conservation equations before substituting any numbers, because the algebra shows which quantity is unknown.