doPhysics · Centre of Mass and Collisions
Collisions
What happens when two bodies strike each other. The stages of a collision and the common velocity. Elastic and inelastic collisions and the coefficient of restitution. Collisions at an angle, from a ball bouncing off a floor to pucks that part at a right angle.
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Why this chapter
Why this chapter exists
A collision is over in a few thousandths of a second, and the forces inside it are huge and hard to follow. Yet the motion before and after can be predicted without them. Momentum survives every collision, because the contact forces are a third-law pair. Kinetic energy may or may not survive, and one number, the coefficient of restitution, says how much of it does. With these two ideas you can solve any head-on collision and find how far a spring bumper squeezes. You can measure the speed of a bullet and follow two balls that meet at an angle.
You will learn
By the end of this chapter you will
- Explain why momentum is conserved in every collision, and find the common velocity at the greatest deformation.
- Find the greatest compression of a spring bumper between two moving blocks.
- Tell elastic, inelastic and completely inelastic collisions apart, and find the kinetic energy lost when bodies stick.
- Use the elastic head-on results and their three limits: equal masses, a heavy striker and a light striker.
- Define the coefficient of restitution and use it with momentum to solve any head-on collision, a bouncing ball and a ballistic pendulum.
- Solve an oblique collision of smooth bodies along the line of impact and the common tangent.
- Use the right-angle rule for equal masses and the geometry of discs of different sizes.
Before you begin
Before you begin
From the chapter on Accelerated Objects, you should know impulse, \(J = F\Delta t = \Delta p\). From the chapters on Work and Energy, you should know kinetic energy \(\tfrac12 mv^2\) and the energy of a spring \(\tfrac12 kx^2\). You should also know the speed \(\sqrt{2gh}\) gained in a fall. You should be able to resolve a vector into two perpendicular components. The chapters on Centre of Mass and on Momentum and Its Conservation help, but this chapter recalls what it needs.
Lessons
1
What Happens in a Collision
A collision is a large force for a very short time. The contact forces are a third-law pair, so the momentum of the pair survives every collision. A head-on collision passes through approach, deformation and restitution, and at the greatest deformation both bodies share one velocity.
A collision is a brief large forceStages of a head on collisionMaximum compression at the common velocity
2
Elastic and Inelastic Collisions
Momentum survives every collision; kinetic energy decides its type. Elastic, inelastic and completely inelastic collisions, and the kinetic energy lost when two bodies stick together.
Kinetic energy sorts collisionsKinetic energy lost when bodies stick
3
Elastic Collisions Head-on
Momentum and kinetic energy together fix both final velocities of an elastic head-on collision. The bodies separate as fast as they approached. Equal masses swap velocities, a heavy body flings a light one ahead, and a light body rebounds from a heavy one.
The elastic head on resultsEqual heavy and light targetsFinding an unknown mass from an elastic collision
4
The Coefficient of Restitution
One number, \(e\), places a collision between sticking and perfectly elastic. With momentum it solves every head-on collision. A ball bounces to \(e^2\) of its height each time, and a bullet in a hanging block measures its own speed.
E is separation over approachTwo equations solve every head on collisionA bouncing ball and e squaredThe ballistic pendulum
5
Oblique Collisions
When bodies meet off-centre, the velocities no longer lie on one line. The line of impact and the common tangent are the natural axes. Momentum is conserved along every direction, a smooth contact leaves each tangential component unchanged, and e acts along the line of impact only.
Line of impact and common tangentMomentum is conserved along every directionThe smooth ball rule and e along the line of impact
6
Right Angles, Floors and Discs
Three oblique collisions that every student meets. A ball bouncing off a smooth floor keeps its speed along the floor. Equal masses, one at rest, leave at a right angle after an elastic collision. Discs of different sizes split at an angle fixed by their radii.
A ball bouncing obliquely off a smooth floorEqual masses split at a right angleDiscs of different sizes
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- ballistic pendulum
- A block hanging from strings that catches a bullet. Momentum through the impact and energy through the swing give the bullet's speed from the height of the swing.
- coefficient of restitution
- The velocity of separation divided by the velocity of approach, measured along the line of impact, \(e = (v_2 - v_1)/(u_1 - u_2)\). It lies between 0 for bodies that stick and 1 for an elastic collision.
- collision
- A brief interaction in which two bodies push on each other with large forces for a very short time, sharply changing their velocities. The total momentum of the pair is the same just before and just after.
- common tangent
- The line through the point of contact at right angles to the line of impact. For smooth bodies no force acts along it, so each body keeps its velocity component along it.
- common velocity
- The single velocity that both bodies share at the instant of greatest deformation, \((m_1u_1 + m_2u_2)/(m_1 + m_2)\). It is the velocity of the centre of mass of the pair.
- completely inelastic collision
- A collision in which the bodies stick together and continue with the common velocity. It loses the most kinetic energy that momentum allows, and \(e = 0\).
- deformation
- The squashing of two bodies at the point of contact while they press together. It stores part of the kinetic energy for a moment.
- elastic collision
- A collision in which the total kinetic energy after the collision equals the total before it. The bodies regain their shape fully, and \(e = 1\).
- head-on collision
- A collision in which all the velocities, before and after, lie along the line joining the centres of the bodies. It is also called a one-dimensional collision.
- impulsive force
- A very large force that acts for a very short time, such as the contact force in a collision. Its effect is measured by its impulse, the area under its force-time graph.
- inelastic collision
- A collision in which some kinetic energy becomes heat, sound or a lasting change of shape. The total kinetic energy after is less than before.
- line of centres
- The line joining the centres of two colliding spheres at the instant of contact. For smooth spheres the contact force acts along it.
- line of impact
- The line at right angles to the two surfaces at the point of contact. For two spheres it passes through both centres. A smooth contact pushes along this line only.
- oblique collision
- A collision in which the velocities do not all lie on one line, such as two balls meeting off-centre. Momentum is then conserved separately along two perpendicular directions.
- reduced mass
- The combination \(m_1m_2/(m_1 + m_2)\) of two masses. The kinetic energy of the relative motion of the pair is half the reduced mass times the square of the relative velocity.
- restitution
- The last stage of a collision, in which the deformed bodies regain some or all of their shape and push each other apart.
- velocity of approach
- The rate at which the gap between two bodies closes before a collision, \(u_1 - u_2\) along the line of centres.
- velocity of separation
- The rate at which the gap between two bodies opens after a collision, \(v_2 - v_1\) along the line of centres.