Exercises: Elastic and Inelastic Collisions

Two carts collide on a track and stick together. Cart A has mass $1.0\ \text{kg}$ and moves at $2.0\ \text{m/s}$; cart B has mass $1.0\ \text{kg}$ and is at rest. Find the speed of the combined carts after the collision.

Classify each collision as elastic, inelastic, or perfectly inelastic, and justify briefly: (a) two steel balls bounce off each other with no loss of total kinetic energy; (b) a lump of clay hits a wall and sticks; (c) two bumper cars collide and bounce apart, with some kinetic energy turned into thermal energy and sound.

A $0.20\ \text{kg}$ ball moving at $5.0\ \text{m/s}$ collides head-on with an identical ball at rest in an elastic collision. State the final velocities of both balls and verify that the total kinetic energy is unchanged.

A $1200\ \text{kg}$ car moving east at $15\ \text{m/s}$ rear-ends a $1800\ \text{kg}$ truck moving east at $10\ \text{m/s}$. After the collision, the two vehicles lock together. (a) Find their common velocity. (b) Calculate the kinetic energy lost in the collision.

A $2.0\ \text{kg}$ cart moving at $3.0\ \text{m/s}$ to the right collides with a $1.0\ \text{kg}$ cart at rest. After the collision, the $2.0\ \text{kg}$ cart moves right at $1.0\ \text{m/s}$ and the $1.0\ \text{kg}$ cart moves right at $4.0\ \text{m/s}$. (a) Verify that momentum is conserved. (b) Determine whether the collision is elastic, and justify your answer with a calculation.

A $0.050\ \text{kg}$ bullet moving at $400\ \text{m/s}$ embeds itself in a $2.0\ \text{kg}$ wooden block hanging at rest from a long string. (a) Find the speed of the block and bullet just after the collision. (b) Find the maximum height the block rises. Use $g = 9.8\ \text{m/s}^2$.

Cart A ($m_A = 0.30\ \text{kg}$) moves right at $2.0\ \text{m/s}$ and collides with cart B ($m_B = 0.90\ \text{kg}$) moving left at $1.0\ \text{m/s}$. The collision is perfectly elastic. (a) Write the two conservation equations you would use. (b) Find the velocity of each cart after the collision. (c) Verify with the kinetic energy that the collision is elastic. (d) Explain, using your answer to (b), how the relative speed of approach compares with the relative speed of separation.

On an air track, a $0.40\ \text{kg}$ glider moving at $1.5\ \text{m/s}$ collides with a $0.60\ \text{kg}$ glider at rest. A student measures the final velocities in two trials. Trial 1: the gliders stick together. Trial 2: the $0.40\ \text{kg}$ glider moves at $-0.30\ \text{m/s}$ and the $0.60\ \text{kg}$ glider at $+1.20\ \text{m/s}$. (a) For each trial, find the final velocities (Trial 1) or check momentum conservation (Trial 2). (b) Compute the fraction of the initial kinetic energy lost in each trial. (c) Classify each trial, and argue which trial involves the larger energy loss and where that energy goes.