a bullet with a mass of 20 g travelling at velocity of 500 m/s strikes a stationary block of mass 3kg. the bullet and the block slides across a rough surface for 1.55 m before rest. Calculate velocity
of what? bullet? block?
when? on impact?
To calculate the final velocity of the bullet and the block, we can use the principle of conservation of momentum.
According to the principle of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.
Momentum (p) is calculated by multiplying the mass (m) by the velocity (v): p = mv.
Given:
- Mass of the bullet (m1) = 20 g = 0.02 kg
- Velocity of the bullet before the collision (v1) = 500 m/s
- Mass of the block (m2) = 3 kg
- Velocity of the block before the collision (v2) = 0 m/s (stationary)
Step 1: Calculate the total momentum before the collision
Momentum before the collision (p_initial) = p1_initial + p2_initial
Where p1_initial = m1 * v1 (momentum of the bullet)
And p2_initial = m2 * v2 (momentum of the block)
p1_initial = 0.02 kg * 500 m/s = 10 kg·m/s
p2_initial = 3 kg * 0 m/s = 0 kg·m/s
So, p_initial = p1_initial + p2_initial = 10 kg·m/s + 0 kg·m/s = 10 kg·m/s
Step 2: Calculate the total momentum after the collision (which is equal to the momentum before the collision due to conservation of momentum)
p_final = p_initial = 10 kg·m/s
Step 3: Calculate the final velocity of the bullet and the block
Momentum after the collision = p_final = m1 * v1_final + m2 * v2_final
Given that the bullet and the block slide across a rough surface before resting, we can assume there is friction, which will bring them to rest. Therefore, the final velocity of both the bullet and the block is 0 m/s.
Hence, 0 = 0.02 kg * v1_final + 3 kg * 0
Solving for v1_final:
0.02 kg * v1_final = 0
v1_final = 0 / 0.02 kg
v1_final = 0 m/s
Therefore, the final velocity of both the bullet and the block is 0 m/s.
To calculate the final velocity, we need to apply the principles of conservation of momentum and Newton's laws of motion. Here's how you can solve this problem step by step:
1. Begin by calculating the initial momentum of the bullet (Pbullet) and the block (Pblock) separately. Momentum is defined as the product of an object's mass and velocity.
Pbullet = mass(bullet) * velocity(bullet) = 0.02 kg * 500 m/s
Pbullet = 10 kg·m/s
Pblock = mass(block) * velocity(block)
Since the block is stationary initially, the initial velocity (velocity(block)) is 0. Therefore, the initial momentum of the block is zero (Pblock = 0).
2. According to the principle of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.
Pinitial = Pfinal
Pbullet + Pblock = Pfinal
Since Pblock = 0 initially, the equation becomes:
Pbullet = Pfinal
3. Now, we need to consider the kinetic energy lost due to the frictional force acting on the bullet-block system. This loss of kinetic energy results in the system finally coming to rest.
4. Apply the work-energy theorem to determine the work done by the frictional force. The work done by the frictional force is equal to the change in kinetic energy.
Frictional force * distance = change in kinetic energy
The change in kinetic energy can be calculated as the difference between the initial kinetic energy and the final kinetic energy.
Initial kinetic energy = 0.5 * mass(system) * velocity(initial)^2
final kinetic energy = 0.5 * mass(system) * velocity(final)^2
Since the block is stationary initially, the initial kinetic energy is zero. Thus, the change in kinetic energy can be simplified to:
Frictional force * distance = final kinetic energy
5. As the system comes to rest, the final kinetic energy becomes zero, so the equation becomes:
Frictional force * distance = 0
Frictional force = 0
6. Since the frictional force is zero, we can conclude that there is no external force acting on the system after the collision. Therefore, the total momentum after the collision (Pfinal) is conserved.
7. Using the principle of conservation of momentum:
Pbullet = Pfinal
Pbullet = mass(bullet) * velocity(final)
8. Rearrange the equation to solve for the final velocity (velocity(final)):
velocity(final) = Pbullet / mass(bullet)
Substituting the known values:
velocity(final) = 10 kg·m/s / 0.02 kg
9. Calculate the final velocity:
velocity(final) = 500 m/s
Thus, the final velocity of the system is 500 m/s.