A heavy wooden block rests on a flat table and a high-speed bullet is fired horizontally into the block, the bullet stopping in it. How far will the block slide before coming to a stop? The mass of the bullet is 10.5 g, the mass of the block is 10.5 kg the bullet’s impact speed is 750 m/s, and the coefficient of kinetic friction between the block and the table is 0.220. (Assume that the bullet does not cause the block to spin.)
This question was re-posted a few hours later, and answered by BobPursley and myself
To find how far the block will slide before coming to a stop, we can apply the principles of conservation of momentum and the work-energy theorem. Let's break down the problem into steps to find the answer.
Step 1: Calculate the initial momentum of the bullet-block system.
The initial momentum of the system is given by the product of the bullet's mass and velocity, as well as the mass of the block (since it is initially at rest).
Initial momentum = (mass of bullet) * (velocity of bullet) + (mass of block) * 0
Initial momentum = (10.5 g) * (750 m/s) + (10.5 kg) * 0
Note: Convert the mass of the bullet to kilograms by dividing by 1000.
Step 2: Calculate the final momentum of the bullet-block system.
Since the bullet stops inside the block, the final momentum of the system is 0, as there is no motion.
Final momentum = 0
Step 3: Use the law of conservation of momentum to find the initial velocity of the block.
According to the law of conservation of momentum, the total momentum before and after the collision should be the same.
Initial momentum = Final momentum
(Initial momentum of the bullet) + (Initial momentum of the block) = 0
(10.5 g) * (750 m/s) + (10.5 kg) * (initial velocity of the block) = 0
Solve for the initial velocity of the block:
(initial velocity of the block) = - (10.5 g * 750 m/s) / (10.5 kg)
Step 4: Use the work-energy theorem to find the distance the block slides.
The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy.
The work done on the block is given by the force of kinetic friction multiplied by the distance it slides:
Work = force of kinetic friction * distance
The force of kinetic friction can be calculated using the coefficient of kinetic friction and the normal force. The normal force acting on the block is equal to its weight:
Normal force = mass of the block * gravitational force
With the normal force, we can calculate the force of kinetic friction:
Force of kinetic friction = coefficient of kinetic friction * normal force
The work done on the block is then equal to the force of kinetic friction multiplied by the distance the block slides:
Work = force of kinetic friction * distance
Using the work-energy theorem, we can equate it to the change in kinetic energy:
Work = change in kinetic energy
Since the block comes to a stop, the final kinetic energy is 0. Therefore:
Work = (initial kinetic energy) - 0
Since the initial kinetic energy of the block is 0.5 * mass * (initial velocity)², we can rewrite the equation:
Force of kinetic friction * distance = 0.5 * mass * (initial velocity)²
Solve for the distance:
Distance = [0.5 * mass * (initial velocity)²] / (force of kinetic friction)
Substituting the values:
Distance = [0.5 * 10.5 kg * [(10.5 g * 750 m/s) / (10.5 kg)]²] / [0.220 * (10.5 kg * 9.8 m/s²)]
Note: Don't forget to convert the mass of the bullet to kilograms.
By calculating this equation, you will find the distance the block slides before coming to a stop.