In a laboratory experiment, a bullet of mass 47.0 g is shot horizontally into a stationary block of totara of mass 3.50 kg that hangs from a light string. The bullet is then embedded a distance d = 3.50 cm into the block and the combined block-bullet object moves at a speed of 3.30 m s¡V1 immediately after impact. Note: g = 9.80 m s¡V2
(a) If instead of being suspended, the block was resting on a rough table, how far would it move before coming to a stop after the bullet's impact? [Assume £gk = 0.413 for the coefficient of kinetic friction applicable to the surface of contact between the block and the table]
To solve this problem, we need to apply the principles of conservation of momentum and conservation of energy. Here's how we can do it step by step:
1. Calculate the initial momentum of the bullet before impact:
- Mass of the bullet = 47.0 g = 0.047 kg
- Initial velocity of the bullet = 0 m/s (since it's stationary)
- Initial momentum = mass x velocity = 0.047 kg x 0 m/s = 0 kg·m/s
2. Calculate the final momentum of the combined block-bullet system after impact:
- Final velocity of the combined system = 3.30 m/s (given)
- Final momentum = (mass of block + mass of bullet) x velocity
- Mass of block = 3.50 kg
- Final momentum = (3.50 kg + 0.047 kg) x 3.30 m/s = 11.6935 kg·m/s
3. Apply the principle of conservation of momentum:
- According to the conservation of momentum, the initial momentum is equal to the final momentum.
- Initial momentum = Final momentum
- 0 kg·m/s = 11.6935 kg·m/s
4. Determine the change in momentum of the block due to the bullet's impact:
- Change in momentum = Final momentum - Initial momentum
- Change in momentum = 11.6935 kg·m/s - 0 kg·m/s = 11.6935 kg·m/s
5. Use the work-energy theorem to calculate the work done by friction:
- The work done by friction is equal to the change in kinetic energy of the block-bullet system.
- Change in kinetic energy = (1/2) x (mass of block + mass of bullet) x (final velocity)^2
- Change in kinetic energy = (1/2) x (3.50 kg + 0.047 kg) x (3.30 m/s)^2
6. Calculate the frictional force:
- The frictional force is equal to the product of the coefficient of kinetic friction (£gk) and the normal force.
- Normal force = mass of block x acceleration due to gravity
- Normal force = 3.50 kg x 9.80 m/s^2
- Frictional force = £gk x (3.50 kg x 9.80 m/s^2)
7. Determine the distance moved by the block on the rough table before coming to a stop:
- The work done by friction is equal to the force of friction multiplied by the distance traveled by the block.
- Work done by friction = Frictional force x distance
- Distance = Work done by friction / Frictional force
By plugging in the known values for £gk, mass of block, final velocity, and solving the equations step by step, we can find the distance moved by the block on the rough table before coming to a stop.