Zaid Berjis

PhysicsLesson 17 of 18About 45 minutes

Elastic and inelastic collisions

When a rugby player tackles an opponent, the two move off as one. When a golf club strikes a ball, the ball springs away. Momentum is conserved in both, but something else is different: what happens to the kinetic energy. This lesson sorts collisions by that difference.

By the end of this lesson you can

  • identify perfectly inelastic, inelastic and elastic collisions;
  • work out how fast objects that stick together move afterward;
  • calculate how much kinetic energy is lost in an inelastic collision;
  • use both momentum and kinetic energy conservation for elastic collisions.

Three kinds of collision

In every collision between isolated objects, momentum is conserved. Kinetic energy is a different story. Some of it can turn into sound, heat, or the energy of bending and denting. Collisions are named by what happens to kinetic energy:

TypeWhat the objects doMomentumKinetic energy
Perfectly inelasticStick together and move as oneConservedNot conserved; the most is lost
InelasticBounce apart, but are deformed or heatedConservedNot conserved; some is lost
ElasticBounce apart with no lasting changeConservedConserved

Most real collisions are inelastic, somewhere between the two ideal cases. Collisions between very hard objects, such as billiard balls or steel marbles, come close to elastic.

Perfectly inelastic collisions

Definition

In a perfectly inelastic collision, the two objects stick together after colliding and move with one common velocity.

Since the objects move together afterward, there is only one unknown final velocity, and conservation of momentum alone finds it:

Perfectly inelastic collision

$$m_1\vec v_{1,i} + m_2\vec v_{2,i} = (m_1 + m_2)\vec v_f$$

Carts that stick together

Simulation

Try this

  1. Press Go! and compare the total momentum before and after in the table.
  2. Change the masses and velocities. Can you find a setting where the joined carts stop dead?

Worked example 1

A 1000 kg car traveling east at 20.0 m/s rear-ends a 1500 kg car stopped at a light. The two cars lock bumpers and slide together. What is their velocity just after the collision?

  1. East is positive. \(m_1 = 1000\ \text{kg}\), \(v_{1,i} = 20.0\ \text{m/s}\); \(m_2 = 1500\ \text{kg}\), \(v_{2,i} = 0\).
  2. Momentum before. \((1000)(20.0) + 0 = 2.00 \times 10^4\ \text{kg·m/s}\)
  3. Momentum after, moving together: \((1000 + 1500)v_f = 2500\,v_f\)
  4. Solve. \(v_f = \dfrac{2.00 \times 10^4}{2500} = 8.00\ \text{m/s}\)

    8.00 m/s east

Worked example 1, simulated

Simulation

Kinetic energy is lost

Think first

Two identical lumps of clay fly at each other with the same speed, hit, and stop dead. Momentum was conserved (it was zero before and after). What happened to their kinetic energy?

Compare your answer

All of it was turned into other forms: mostly thermal energy and the work of squashing the clay out of shape, plus a little sound. Momentum is conserved; kinetic energy is not.

Objects that stick together always lose some kinetic energy to other forms. Compare the total kinetic energy at the start with the total at the end to see how much:

$$\Delta KE = KE_f - KE_i$$

Where does the energy go?

Simulation

Try this

Watch the energy bar during the collision. Orange is kinetic energy, yellow is energy stored in the squashed bumpers, and the dashed part is energy lost. How much kinetic energy is left when two equal carts with equal speeds stick?

Worked example 2

A 0.400 kg ball of clay moving at 5.00 m/s to the right hits a 0.200 kg ball of clay moving at 2.00 m/s to the left. They stick together. (a) Find their final velocity. (b) How much kinetic energy is lost?

  1. Right is positive. \(v_{1,i} = +5.00\ \text{m/s}\), \(v_{2,i} = -2.00\ \text{m/s}\).
  2. (a) Momentum. \((0.400)(5.00) + (0.200)(-2.00) = 2.00 - 0.400 = 1.60\ \text{kg·m/s}\). So \(v_f = \dfrac{1.60}{0.600} = 2.67\ \text{m/s}\) to the right.
  3. (b) Kinetic energy before. \(\tfrac12(0.400)(5.00)^2 + \tfrac12(0.200)(2.00)^2 = 5.00 + 0.400 = 5.40\ \text{J}\)
  4. Kinetic energy after. \(\tfrac12(0.600)(2.67)^2 = 2.13\ \text{J}\)
  5. Change. \(\Delta KE = 2.13 - 5.40 = -3.27\ \text{J}\)

    (a) 2.67 m/s to the right. (b) 3.27 J of kinetic energy becomes other forms.

Worked example 2, simulated

Simulation

Elastic collisions

Definition

In an elastic collision, the objects bounce apart and the total kinetic energy after the collision equals the total before.

During an elastic collision the objects do squash a little, storing energy like springs, but they give all of it back as they separate. In an elastic collision two equations hold at once:

Elastic collision

$$m_1 v_{1,i} + m_2 v_{2,i} = m_1 v_{1,f} + m_2 v_{2,f}$$ $$\tfrac12 m_1 v_{1,i}^2 + \tfrac12 m_2 v_{2,i}^2 = \tfrac12 m_1 v_{1,f}^2 + \tfrac12 m_2 v_{2,f}^2$$

Stick, lose some energy, or keep it all

Simulation

Try this

  1. Run the same carts with each of the three bumper types. Compare total momentum and total kinetic energy after each.
  2. With equal masses and the elastic bumper, what happens to the two velocities?

Equal balls, elastic collision

Simulation

Try this

Two equal balls collide elastically head-on. Compare their velocities before and after. Try a few different speeds: what always happens?

Worked example 3

A 20.0 g steel ball rolling at 0.300 m/s to the right hits a 40.0 g steel ball at rest, head-on, in an elastic collision. Afterward the small ball moves at 0.100 m/s to the left. (a) Find the velocity of the large ball. (b) Check that kinetic energy is conserved.

  1. Right is positive. \(v_{1,i} = 0.300\ \text{m/s}\), \(v_{2,i} = 0\), \(v_{1,f} = -0.100\ \text{m/s}\). Masses in kilograms: 0.0200 and 0.0400.
  2. (a) Momentum. \((0.0200)(0.300) + 0 = (0.0200)(-0.100) + (0.0400)v_{2,f}\), so \(0.00600 = -0.00200 + 0.0400\,v_{2,f}\), and \(v_{2,f} = 0.200\ \text{m/s}\) to the right.
  3. (b) KE before. \(\tfrac12(0.0200)(0.300)^2 = 9.00 \times 10^{-4}\ \text{J}\)
  4. KE after. \(\tfrac12(0.0200)(0.100)^2 + \tfrac12(0.0400)(0.200)^2 = 1.00 \times 10^{-4} + 8.00 \times 10^{-4} = 9.00 \times 10^{-4}\ \text{J}\)

    (a) 0.200 m/s to the right. (b) Kinetic energy is \(9.00 \times 10^{-4}\ \text{J}\) before and after, so the collision is elastic.

Worked example 3, simulated

Simulation

Real bounces

Drop a superball, a tennis ball and a ball of clay onto a hard floor. Each collides with Earth, which is so massive it barely moves. How high each ball bounces back shows how close the collision is to elastic.

Three balls dropped on a hard floor

Simulation

Try this

Predict the order of the bounce heights, then drop all three. Which collision is closest to elastic, and which is closest to perfectly inelastic?

Check yourself

Question 1

A 1200 kg car moving at 25.0 m/s rear-ends an 1800 kg van moving at 15.0 m/s in the same direction. They lock together. How fast do they move just after the collision?

Show solution

\(v_f = \dfrac{(1200)(25.0) + (1800)(15.0)}{3000} = \dfrac{30\,000 + 27\,000}{3000} = 19.0\ \text{m/s}\)

Question 2

How much kinetic energy is lost in the collision in Question 1?

Show solution

Before: \(\tfrac12(1200)(25.0)^2 + \tfrac12(1800)(15.0)^2 = 375\,000 + 202\,500 = 577\,500\ \text{J}\).

After: \(\tfrac12(3000)(19.0)^2 = 541\,500\ \text{J}\). Lost: \(3.60 \times 10^4\ \text{J}\). Little is lost because the vehicles were moving at similar speeds.

Question 3

A 50.0 g lump of putty moving at 8.00 m/s hits a 200 g cart at rest on a frictionless track and sticks to it. How fast does the cart move afterward?

Show solution

\(v_f = \dfrac{(0.0500)(8.00)}{0.250} = 1.60\ \text{m/s}\). Kinetic energy drops from 1.60 J to 0.320 J: 80% is lost.

Question 4

A moving pool ball hits an identical ball at rest head-on, in a nearly elastic collision. What happens?

  1. Both balls move forward at half the original speed.
  2. The first ball stops and the second moves off at the first ball's original speed.
  3. The first ball bounces straight back at its original speed.
  4. Both balls stop.
Show solution

For equal masses in an elastic collision, the velocities swap. Only this answer conserves both momentum and kinetic energy. (Answer A conserves momentum but loses half the kinetic energy.)

Question 5

Two cars collide, crumple, and separate. Which kind of collision is this?

  1. Elastic
  2. Perfectly inelastic
  3. Inelastic
  4. It cannot be classified
Show solution

They bounce apart, so it is not perfectly inelastic. The crumpling means kinetic energy went into permanently deforming the metal, so it is not elastic either.

Summary

  • Momentum is conserved in every collision between isolated objects.
  • Perfectly inelastic: objects stick together; \(m_1v_{1,i} + m_2v_{2,i} = (m_1 + m_2)v_f\). Kinetic energy is lost.
  • Inelastic: objects bounce apart, but some kinetic energy becomes other forms.
  • Elastic: both momentum and kinetic energy are conserved. Equal masses in a head-on elastic collision swap velocities.

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