A bullet of mass 5 g with velocity 1000m/s hits a wooden block of 1.5kg at rest vertically. 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 find the maximum height in which the block is lifted, we need to consider the conservation of momentum and energy.

Step 1: Calculate the momentum of the bullet before impact:
Momentum = mass × velocity
Momentum of the bullet = (5 g) × (1000 m/s)
= (5/1000) kg × 1000 m/s
= 5 kg⋅m/s

Step 2: Calculate the momentum of the bullet after impact:
Momentum of the bullet = (5 g) × (400 m/s)
= (5/1000) kg × 400 m/s
= 2 kg⋅m/s

Step 3: Calculate the momentum transferred to the block:
Momentum transferred to the block = Momentum of the bullet before impact - Momentum of the bullet after impact
= 5 kg⋅m/s - 2 kg⋅m/s
= 3 kg⋅m/s

Step 4: Calculate the velocity of the block immediately after impact using the conservation of momentum:
Momentum of the block = mass of the block × velocity of the block
3 kg⋅m/s = 1.5 kg × velocity of the block
velocity of the block = 3 kg⋅m/s / 1.5 kg
= 2 m/s

Step 5: Calculate the change in kinetic energy of the block:
Change in kinetic energy = (1/2) × mass × (final velocity)^2 - (1/2) × mass × (initial velocity)^2
Change in kinetic energy = (1/2) × 1.5 kg × (2 m/s)^2 - (1/2) × 1.5 kg × (0 m/s)^2
= 1.5 kg × (2 m/s)^2 - 0
= 1.5 kg × 4 m^2/s^2
= 6 J

Step 6: Calculate the potential energy gained by the block due to its vertical displacement:
Potential energy = mass × gravity × height
6 J = 1.5 kg × 9.8 m/s^2 × height
height = 6 J / (1.5 kg × 9.8 m/s^2)
= 6 / (1.5 × 9.8) m
≈ 0.408 m

Therefore, the maximum height in which the block is lifted from its initial position is approximately 0.408 meters.

To solve this problem, we can use the principle of conservation of momentum.

The initial momentum of the bullet (before impact) is given by the product of its mass and velocity:

Initial momentum of bullet = mass of bullet × initial velocity of bullet

= 5 g × 1000 m/s (converting grams to kilograms)

= 0.005 kg × 1000 m/s

= 5 kg·m/s

Since the block is at rest initially, its initial momentum is zero:

Initial momentum of block = 0

According to the conservation of momentum, the total momentum after the collision is equal to the total momentum before the collision:

Total momentum after collision = Total momentum before collision

The final momentum of the bullet and the block can be calculated as follows:

Final momentum of bullet = mass of bullet × final velocity of bullet

= 5 g × 400 m/s (converting grams to kilograms)

= 0.005 kg × 400 m/s

= 2 kg·m/s

Final momentum of block = mass of block × final velocity of block

= 1.5 kg × 0 m/s (since the block is lifted and comes to rest)

= 0 kg·m/s

Now, substituting these values into the conservation of momentum equation:

Final momentum of bullet + Final momentum of block = Initial momentum of bullet + Initial momentum of block

2 kg·m/s + 0 kg·m/s = 5 kg·m/s + 0

2 kg·m/s = 5 kg·m/s

To find the maximum height the block is lifted, we can use the principle of conservation of mechanical energy.

The initial mechanical energy of the system (bullet + block) is given by the sum of the kinetic energy of the bullet and the potential energy of the block:

Initial mechanical energy = 0.5 × mass of bullet × (initial velocity of bullet)^2 + mass of block × g × initial height

where g is the acceleration due to gravity and initial height is 0 since the block is at ground level.

The final mechanical energy is given by the sum of the kinetic energy of the bullet (which is 0.5 × mass of bullet × (final velocity of bullet)^2) and the potential energy of the block at its maximum height (mass of block × g × maximum height).

Since there is no external force doing work on the system, we can equate the initial mechanical energy to the final mechanical energy:

0.5 × mass of bullet × (initial velocity of bullet)^2 + mass of block × g × initial height = 0.5 × mass of bullet × (final velocity of bullet)^2 + mass of block × g × maximum height

Plugging in the known values:

0.5 × 0.005 kg × (1000 m/s)^2 + 1.5 kg × 9.8 m/s^2 × 0 = 0.5 × 0.005 kg × (400 m/s)^2 + 1.5 kg × 9.8 m/s^2 × maximum height

Simplifying the equation:

0.5 × 0.005 kg × (1000 m/s)^2 = 0.5 × 0.005 kg × (400 m/s)^2 + 1.5 kg × 9.8 m/s^2 × maximum height

2500 J = 400 J + 14.7 J/kg × maximum height

Subtracting 400 J from both sides:

2100 J = 14.7 J/kg × maximum height

Dividing by 14.7 J/kg:

maximum height = 2100 J / 14.7 J/kg

maximum height ≈ 142.86 kg

Therefore, the maximum height to which the block is lifted from its initial position is approximately 142.86 kg.