A bullet of mass m = 0.01 kg and velocity v0 = 21 m/s is fired toward a block of mass 4m. The block is initially at rest on a frictionless horizontal surface.
The bullet penetrates the block and emerges with a velocity of 7 m/s.
(a) Determine the final speed of the block. (3) (b) Determine the loss in kinetic energy of the bullet. (3)
(c) Determine the gain in the kinetic energy of the block. Then calculate the overall loss in kinetic energy. What form of energy is this energy loss? (6)
(a) conserve momentum.
0.01*21 = (0.01+4)v (assuming a "mass 4m" is really 4 kg)
(b,c) KE = 1/2 mv^2, so ...
To solve these questions, we need to apply the principles of conservation of momentum and conservation of kinetic energy.
(a) Determine the final speed of the block:
We can start by applying the principle of conservation of momentum. The total momentum before the collision should be equal to the total momentum after the collision.
The initial momentum of the bullet is given by:
Initial momentum of bullet = mass of bullet × initial velocity of bullet
= m × v0
The initial momentum of the block is zero since it is initially at rest.
The final momentum of the bullet is given by:
Final momentum of bullet = mass of bullet × final velocity of bullet
= m × vf (since vf is the final velocity of the bullet)
The final momentum of the block is given by:
Final momentum of block = mass of block × final velocity of block
Applying the conservation of momentum:
Initial momentum of bullet = Final momentum of bullet + Final momentum of block
m × v0 = m × vf + 4m × vf (substituting for the momenta)
Simplifying the equation:
v0 = vf + 4vf
v0 = 5vf
Now, we substitute the given values for v0 and solve for vf:
21 m/s = 5vf
Finally, we find:
vf = 21 m/s / 5
vf = 4.2 m/s
Therefore, the final speed of the block is 4.2 m/s.
(b) Determine the loss in kinetic energy of the bullet:
The loss in kinetic energy of the bullet can be determined by calculating the change in kinetic energy of the bullet before and after the collision.
The initial kinetic energy of the bullet is given by:
Initial kinetic energy of bullet = (1/2) × mass of bullet × (initial velocity of bullet)^2
= (1/2) × m × (v0)^2
The final kinetic energy of the bullet is given by:
Final kinetic energy of bullet = (1/2) × mass of bullet × (final velocity of bullet)^2
= (1/2) × m × (vf)^2
The loss in kinetic energy of the bullet is then calculated as the difference between the initial and final kinetic energies:
Loss in kinetic energy of bullet = Initial kinetic energy of bullet - Final kinetic energy of bullet
Substituting the given values:
Loss in kinetic energy of bullet = (1/2) × m × (v0)^2 - (1/2) × m × (vf)^2
Calculating the values:
Loss in kinetic energy of bullet = (1/2) × 0.01 kg × (21 m/s)^2 - (1/2) × 0.01 kg × (7 m/s)^2
Simplifying the equation further gives the answer.
(c) Determine the gain in the kinetic energy of the block. Then calculate the overall loss in kinetic energy. What form of energy is this energy loss?
The gain in the kinetic energy of the block can be determined by subtracting the initial kinetic energy of the block from the final kinetic energy of the block.
The initial kinetic energy of the block is zero since it is initially at rest.
The final kinetic energy of the block is given by:
Final kinetic energy of block = (1/2) × mass of block × (final velocity of block)^2
= (1/2) × 4m × (vf)^2
The gain in kinetic energy of the block is then calculated as:
Gain in kinetic energy of block = Final kinetic energy of block - Initial kinetic energy of block
Substituting the given values:
Gain in kinetic energy of block = (1/2) × 4m × (vf)^2 - 0
Calculating the values gives the answer.
The overall loss in kinetic energy is the sum of the loss in the bullet's kinetic energy and the gain in the block's kinetic energy. This energy loss is in the form of thermal energy, or heat, due to the collision and deformation of the bullet and block.
To solve this problem, we can use the principle of conservation of momentum and the principle of conservation of kinetic energy. Let's break down the problem step-by-step:
Step 1: Find the final velocity of the block.
Since the bullet is penetrating the block, we can treat the block and the bullet as a completely closed system.
According to the principle of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.
Initial momentum = Final momentum
(mass of bullet x initial velocity of bullet) + (mass of block x initial velocity of block) = (mass of bullet x final velocity of bullet) + (mass of block x final velocity of block)
(0.01 kg x 21 m/s) + (4m x 0) = (0.01 kg x 7 m/s) + (4m x final velocity of block)
0.21 kg·m/s + 0 = 0.07 kg·m/s + 4m x final velocity of block
To find the final velocity of the block, we can isolate the term:
final velocity of block = (0.21 kg·m/s - 0.07 kg·m/s) / (4m)
final velocity of block = 0.14 kg·m/s / 4m
final velocity of block = 0.035 m/s
Therefore, the final speed of the block is 0.035 m/s.
Step 2: Find the loss in kinetic energy of the bullet.
The initial kinetic energy of the bullet can be calculated using the formula:
Initial kinetic energy of bullet = 0.5 x mass of bullet x (initial velocity of bullet)^2
Initial kinetic energy of bullet = 0.5 x 0.01 kg x (21 m/s)^2
Initial kinetic energy of bullet = 2.205 J
The final kinetic energy of the bullet can be calculated using the formula:
Final kinetic energy of bullet = 0.5 x mass of bullet x (final velocity of bullet)^2
Final kinetic energy of bullet = 0.5 x 0.01 kg x (7 m/s)^2
Final kinetic energy of bullet = 0.245 J
Therefore, the loss in kinetic energy of the bullet is:
Loss in kinetic energy = Initial kinetic energy - Final kinetic energy
Loss in kinetic energy = 2.205 J - 0.245 J
Loss in kinetic energy = 1.96 J
Step 3: Find the gain in kinetic energy of the block.
The initial kinetic energy of the block is zero since it is initially at rest.
The final kinetic energy of the block can be calculated using the formula:
Final kinetic energy of block = 0.5 x mass of block x (final velocity of block)^2
Final kinetic energy of block = 0.5 x 4m x (0.035 m/s)^2
Final kinetic energy of block = 0.00245 J
Therefore, the gain in kinetic energy of the block is 0.00245 J.
Overall loss in kinetic energy = Loss in kinetic energy of bullet + Gain in kinetic energy of block
Overall loss in kinetic energy = 1.96 J + 0.00245 J
Overall loss in kinetic energy = 1.96245 J
The overall loss in kinetic energy is 1.96245 J.
This energy loss is in the form of heat, as it is dissipated during the collision.