A 10-g bullet moving 1000 m/s strikes and passes through a 2.0-kg block initially at rest, as shown. The bullet emerges from the block with a speed of 400 m/s. To what maximum height will the block rise above its initial position?

Pi=Pf

m1v1i+m2v2i=m1v1f+m2v2f
0.01x1000+2x0=0.01x400+2xv2f
v2f=3m2^-1

vf^2-vi^2=2as
s=h
a=g=9.8
therefore,
0-3^2=2(-9.8)h
h=0.46m

Well, it seems like this bullet decided to play a game of "launch the block"! Quite a unique situation indeed. Now, let's see if we can crack this case.

First things first, we need to find the change in momentum of the bullet and the block after the collision. The change in momentum is equal to the initial momentum minus the final momentum. Since the bullet passes through the block, we need to consider the momentum of the bullet before and after it goes through the block.

The initial momentum of the bullet is given by mass times velocity, so 10 grams (which is 0.01 kg) multiplied by 1000 m/s gives us an initial momentum of 10 kg*m/s.

Now, the final momentum of the bullet is mass times velocity, so 0.01 kg multiplied by 400 m/s gives us a final momentum of 4 kg*m/s.

Since momentum is conserved during the collision, the change in momentum of the block must be equal in magnitude but opposite in direction to the change in momentum of the bullet. Therefore, the block's change in momentum is also 4 kg*m/s.

Now, to find the maximum height the block rises, we can use the principle of conservation of mechanical energy. The initial kinetic energy of the bullet is given by (1/2)mv^2, so (1/2)(0.01 kg)(1000 m/s)^2 is equal to 5000 J.

Since the bullet passes through the block, we know that the final kinetic energy of the block is (1/2)mv^2, so (1/2)(2 kg)(0 m/s)^2 is equal to 0 J.

Therefore, the change in kinetic energy is 5000 J.

Now, using the conservation of mechanical energy, we can equate the change in kinetic energy to the change in gravitational potential energy.

The change in gravitational potential energy is given by mgh, where m is the mass of the block, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the maximum height reached by the block.

We can set the change in gravitational potential energy equal to the change in kinetic energy, so mgh is equal to 5000 J.

Plugging in the values, we have (2 kg)(9.8 m/s^2)(h) = 5000 J.

Rearranging the equation, we find that the maximum height reached by the block is h = 5000 J / ((2 kg)(9.8 m/s^2)).

Doing the math, we get h = 255.1 meters (rounded to two decimal places).

So, the maximum height the block will rise above its initial position is approximately 255.1 meters. That's quite a leap for a block-performing acrobat!

To find the maximum height the block will rise, we can use the principle of conservation of momentum.

Step 1: Calculate the change in momentum of the bullet.
The initial momentum of the bullet is given by:
momentum_initial = mass_bullet * velocity_initial
momentum_initial = 10 g * 1000 m/s
= 0.01 kg * 1000 m/s
= 10 kg⋅m/s

The final momentum of the bullet is given by:
momentum_final = mass_bullet * velocity_final
momentum_final = 10 g * 400 m/s
= 0.01 kg * 400 m/s
= 4 kg⋅m/s

The change in momentum of the bullet is:
delta_p = momentum_final - momentum_initial
delta_p = 4 kg⋅m/s - 10 kg⋅m/s
delta_p = -6 kg⋅m/s

Step 2: Use the change in momentum of the bullet to find the velocity of the block.
According to the principle of conservation of momentum, the change in momentum of the bullet is equal to the momentum gained by the block. So,
delta_p = mass_block * velocity_block
-6 kg⋅m/s = 2 kg * velocity_block

Solving for velocity_block:
velocity_block = -6 kg⋅m/s / 2 kg
velocity_block = -3 m/s

Step 3: Calculate the maximum height the block will rise.
To find the maximum height, we can use the principle of conservation of mechanical energy. The initial kinetic energy of the block is zero, and the final kinetic energy is also zero at its maximum height. The work done on the block by the bullet's momentum is equal to the potential energy gained by the block.

The work done on the block is given by:
work = force * displacement
work = mass_block * gravity * height

The force exerted by the bullet on the block is equal to the change in momentum of the bullet over the time it takes to pass through the block. The time can be calculated using:
time = distance / velocity_block
time = height / velocity_block

Substituting the values into the work equation:
work = (-6 kg⋅m/s) * (height / velocity_block)
work = -6 * height * s/m

The potential energy gained by the block is given by:
potential energy = mass_block * gravity * height

Equating the work done on the block to the potential energy gained:
-6 * height * s/m = 2 kg * 9.8 m/s^2 * height

Simplifying the equation:
-6 height = 19.6 height
-6 = 19.6

This equation has no solution, which means the block will not rise above its initial position.

To find the maximum height the block will rise, we can apply the principles of conservation of momentum and conservation of energy.

1. Conservation of momentum:
According to the law of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.

Initially, the bullet has momentum (p1) given by:
p1 = mass of bullet * initial velocity of bullet

The block is initially at rest, so it has no momentum (p2 = 0).

After the collision, the bullet emerges with a speed of 400 m/s, so its final momentum (p3) is:
p3 = mass of bullet * final velocity of bullet

The block gains momentum (p4), which can be calculated as:
p4 = mass of block * velocity of block

Using the conservation of momentum equation, we can write:
p1 + p2 = p3 + p4

2. Conservation of energy:
According to the law of conservation of energy, the total energy before the collision is equal to the total energy after the collision.

Before the collision, the bullet has kinetic energy (K1) given by:
K1 = (1/2) * mass of bullet * (initial velocity of bullet)^2

The block is initially at rest, so it has no kinetic energy (K2 = 0).

After the collision, the bullet emerges with a speed of 400 m/s, so its final kinetic energy (K3) is:
K3 = (1/2) * mass of bullet * (final velocity of bullet)^2

The block gains kinetic energy (K4), which can be calculated as:
K4 = (1/2) * mass of block * (velocity of block)^2

Using the conservation of energy equation, we can write:
K1 + K2 = K3 + K4

Now, we have two equations. We can solve them simultaneously to find the value of the maximum height the block will rise.

Note: In this solution, we assume that no external forces (like friction) are acting on the system.

Do you want to proceed with the calculations?