Section 4.8: Review Examples
This section provides a set of fully worked examples that bring together the concepts of impulse, momentum, and collisions. Review examples are meant to show the problem-solving process step by step, preparing you for more advanced problems and tests.
Example 1: Impulse and Force
A 0.2 kg soccer ball initially at rest is kicked and leaves the foot with a velocity of 15 m/s. If the foot was in contact with the ball for 0.05 s, find the average force exerted on the ball.
Impulse: \( J = \Delta p = m v = 0.2 \times 15 = 3 \, \text{N·s} \)
Average force: \( F = \dfrac{J}{\Delta t} = \dfrac{3}{0.05} = 60 \, \text{N} \)
The average force on the ball is 60 N.
Example 2: Elastic Collision
A 1.0 kg ball moving east at 6 m/s collides elastically with a 2.0 kg ball at rest. Find the velocities of both balls after the collision.
Using conservation of momentum and kinetic energy:
Momentum: \( (1)(6) = (1)v_1 + (2)v_2 \)
Energy: \( \tfrac{1}{2}(1)(6^2) = \tfrac{1}{2}(1)v_1^2 + \tfrac{1}{2}(2)v_2^2 \)
Solving the system:
\( v_1 = -2 \, \text{m/s} \), \( v_2 = 4 \, \text{m/s} \).
The first ball bounces back at 2 m/s, while the second moves east at 4 m/s.
Example 3: Inelastic Collision
A 1200 kg car moving at 10 m/s east collides with an 800 kg car moving at 15 m/s west. The cars stick together after collision. Find the final velocity of the wreck.
Total momentum: \( (1200)(10) + (800)(-15) = 12000 - 12000 = 0 \)
Final mass: \( 2000 \, \text{kg} \)
Final velocity: \( v = \dfrac{0}{2000} = 0 \, \text{m/s} \)
The wreck remains at rest after the collision.
Example 4: Explosion
A firecracker of mass 0.5 kg at rest explodes into two fragments of masses 0.3 kg and 0.2 kg. If the 0.3 kg piece moves at 12 m/s east, find the velocity of the 0.2 kg piece.
Momentum conservation: \( 0 = (0.3)(12) + (0.2)v \)
\( 0 = 3.6 + 0.2v \) → \( v = -18 \, \text{m/s} \)
The smaller fragment moves west at 18 m/s.