# 8.5 Inelastic collisions in one dimension  (Page 3/7)

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## Calculating final velocity and energy release: two carts collide

In the collision pictured in [link] , two carts collide inelastically. Cart 1 (denoted ${m}_{1}$ carries a spring which is initially compressed. During the collision, the spring releases its potential energy and converts it to internal kinetic energy. The mass of cart 1 and the spring is 0.350 kg, and the cart and the spring together have an initial velocity of $2\text{.}\text{00 m/s}$ . Cart 2 (denoted ${m}_{2}$ in [link] ) has a mass of 0.500 kg and an initial velocity of $-0\text{.}\text{500 m/s}$ . After the collision, cart 1 is observed to recoil with a velocity of $-4\text{.}\text{00 m/s}$ . (a) What is the final velocity of cart 2? (b) How much energy was released by the spring (assuming all of it was converted into internal kinetic energy)?

Strategy

We can use conservation of momentum to find the final velocity of cart 2, because ${F}_{\text{net}}=0$ (the track is frictionless and the force of the spring is internal). Once this velocity is determined, we can compare the internal kinetic energy before and after the collision to see how much energy was released by the spring.

Solution for (a)

As before, the equation for conservation of momentum in a two-object system is

${m}_{1}{v}_{1}+{m}_{2}{v}_{2}={m}_{1}{v\prime }_{1}+{m}_{2}{v\prime }_{2}\text{.}$

The only unknown in this equation is ${v\prime }_{2}$ . Solving for ${v\prime }_{2}$ and substituting known values into the previous equation yields

$\begin{array}{lll}{v\prime }_{2}& =& \frac{{m}_{1}{v}_{1}+{m}_{2}{v}_{2}-{m}_{1}{v\prime }_{1}}{{m}_{2}}\\ & =& \frac{\left(\text{0.350 kg}\right)\left(\text{2.00 m/s}\right)+\left(\text{0.500 kg}\right)\left(-\text{0.500 m/s}\right)}{\text{0.500 kg}}-\frac{\left(\text{0.350 kg}\right)\left(-\text{4.00 m/s}\right)}{\text{0.500 kg}}\\ & =& \text{3.70 m/s.}\end{array}$

Solution for (b)

The internal kinetic energy before the collision is

$\begin{array}{lll}{\text{KE}}_{\text{int}}& =& \frac{1}{2}{m}_{1}{v}_{1}^{2}+\frac{1}{2}{m}_{2}{v}_{2}^{2}\\ & =& \frac{1}{2}\left(0\text{.}\text{350 kg}\right){\left(\text{2.00 m/s}\right)}^{2}+\frac{1}{2}\left(0\text{.}\text{500 kg}\right){\left(–0\text{.}\text{500 m/s}\right)}^{2}\\ & =& 0\text{.}\text{763 J}.\end{array}$

After the collision, the internal kinetic energy is

$\begin{array}{lll}{\text{KE}\prime }_{\text{int}}& =& \frac{1}{2}{m}_{1}{v\prime }_{1}^{2}+\frac{1}{2}{m}_{2}{v\prime }_{2}^{2}\\ & =& \frac{1}{2}\left(\text{0.350 kg}\right){\left(-\text{4.00 m/s}\right)}^{2}+\frac{1}{2}\left(\text{0.500 kg}\right){\left(\text{3.70 m/s}\right)}^{2}\\ & =& \text{6.22 J.}\end{array}$

The change in internal kinetic energy is thus

$\begin{array}{lll}{KE\prime }_{\text{int}}-{\text{KE}}_{\text{int}}& =& \text{6.22 J}-0\text{.}\text{763 J}\\ & =& \text{5.46 J.}\end{array}$

Discussion

The final velocity of cart 2 is large and positive, meaning that it is moving to the right after the collision. The internal kinetic energy in this collision increases by 5.46 J. That energy was released by the spring.

## Section summary

• An inelastic collision is one in which the internal kinetic energy changes (it is not conserved).
• A collision in which the objects stick together is sometimes called perfectly inelastic because it reduces internal kinetic energy more than does any other type of inelastic collision.
• Sports science and technologies also use physics concepts such as momentum and rotational motion and vibrations.

## Conceptual questions

What is an inelastic collision? What is a perfectly inelastic collision?

Mixed-pair ice skaters performing in a show are standing motionless at arms length just before starting a routine. They reach out, clasp hands, and pull themselves together by only using their arms. Assuming there is no friction between the blades of their skates and the ice, what is their velocity after their bodies meet?

A small pickup truck that has a camper shell slowly coasts toward a red light with negligible friction. Two dogs in the back of the truck are moving and making various inelastic collisions with each other and the walls. What is the effect of the dogs on the motion of the center of mass of the system (truck plus entire load)? What is their effect on the motion of the truck?

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