A bullet of mass 5 g with velocity 1000m/s hits a wooden block of 1.5kg at rest. If the bullet penetrates through the block with a velocity of 400m/s, calculate the maximum height in which the block is lifted from its initial position.
To calculate the maximum height the block is lifted, we will apply the principle of conservation of momentum and the principle of conservation of kinetic energy.
First, let's calculate the initial momentum of the bullet:
Initial momentum of the bullet = mass x velocity
= 5 g * 1000 m/s
= 5 x 10^(-3) kg * 1000 m/s
= 5 kg m/s
Next, let's calculate the final momentum of the bullet and the block together:
Final momentum = (mass of bullet + mass of block) x final velocity
= (5 g + 1.5 kg) * 400 m/s
= (5 x 10^(-3) kg + 1.5 kg) * 400 m/s
= 1.505 kg * 400 m/s
= 602 kg m/s
Now, let's apply the principle of conservation of momentum:
Initial momentum = Final momentum
5 kg m/s = 602 kg m/s
Therefore, the bullet loses momentum equal to (602 - 5) kg m/s to the wooden block. This loss in momentum is equal to the gain in momentum by the wooden block.
Now, let's solve for the change in velocity of the wooden block:
Change in velocity = Loss in momentum / mass of the block
= (602 kg m/s - 5 kg m/s) / 1.5 kg
= 397 kg m/s / 1.5 kg
= 265.33 m/s
Next, let's calculate the work done on the block to lift it:
Work done = change in kinetic energy
= 1/2 * mass * (final velocity^2 - initial velocity^2)
= 1/2 * 1.5 kg * (0 - 265.33 m/s)^2
= 1/2 * 1.5 kg * (- 265.33 m/s)^2
= 1/2 * 1.5 kg * 70438.72 m^2/s^2
= 52,829.04 J
Finally, let's calculate the maximum height using the work-energy principle:
Work done = mgh (where m is mass, g is acceleration due to gravity, and h is the maximum height)
52,829.04 J = 1.5 kg * 9.8 m/s^2 * h
Therefore, the maximum height to which the block is lifted is:
h = 52,829.04 J / (1.5 kg * 9.8 m/s^2)
= 3593.31 m
To calculate the maximum height to which the block is lifted, we can use the principle of conservation of momentum and the work-energy principle.
Step 1: Calculate the initial momentum of the bullet:
Momentum (p) = mass (m) x velocity (v)
Given: mass (m) of the bullet = 5 g = 0.005 kg
velocity (v) of the bullet = 1000 m/s
Initial momentum of the bullet = 0.005 kg x 1000 m/s = 5 kg*m/s
Step 2: Calculate the final momentum of the bullet after it penetrates the block:
Given: velocity (v) of the bullet after penetration = 400 m/s
Final momentum of the bullet = 0.005 kg x 400 m/s = 2 kg*m/s
Step 3: Calculate the change in momentum:
Change in momentum (Δp) = Final momentum - Initial momentum
Δp = 2 kg*m/s - 5 kg*m/s = -3 kg*m/s
Note: The negative sign indicates a decrease in momentum due to the direction of the momentum change.
Step 4: Calculate the work done on the block:
The work done is equal to the change in kinetic energy of the bullet, which is equal to 1/2 * mass * (final velocity)^2 - 1/2 * mass * (initial velocity)^2.
work done = 1/2 * 0.005 kg * (400 m/s)^2 - 1/2 * 0.005 kg * (1000 m/s)^2
work done = -375 Joules
Here, the negative sign indicates that work was done against the block, resulting in a decrease in kinetic energy.
Step 5: Calculate the maximum height the block is lifted:
The work done on the block is equal to the potential energy gained by the block, given by the formula: potential energy (PE) = mass of block (m) * g * h
Given: mass of block (m) = 1.5 kg
acceleration due to gravity (g) = 9.8 m/s^2
Potential energy gained by the block = 1.5 kg * 9.8 m/s^2 * h
Equating the work done to the potential energy gained:
-375 Joules = 1.5 kg * 9.8 m/s^2 * h
Solving for h:
h = -375 J / (1.5 kg * 9.8 m/s^2)
h ≈ -25.5 meters
Note: The negative sign here indicates that the block is lifted in the opposite direction of gravity, which means it is moving upwards.
Therefore, the maximum height the block is lifted from its initial position is approximately 25.5 meters above the starting point.