A bullet of mass 10g travelling horizontally at a speed of 100m/s embeds itself in a block of wood of mass 990g suspended by strings so that it can swing freely find:
a.the vertical height through which the block rises.
B.how much of the bullet energy becomes internal energy (g
The law of conservation of linear momentum
mv=(m+M)u
The law of conservation of energy
u =mv/(m+M) = 0.01•100/(0.99+0.01)= 1 m/s.
(m+M)u²/2 =(m+M)gh
h= u²/2•g=1/2•9.8=0.05 m
mv²/2- (m+M)u²/2 = 0.01•10000/2 - 1•1/2 = 50-0.5=49.5 J
To find the vertical height through which the block rises, we can use the principle of conservation of mechanical energy. The initial kinetic energy of the bullet is converted into gravitational potential energy of the block.
a. Find the initial kinetic energy of the bullet:
Given:
Mass of the bullet, m = 10 g = 0.01 kg
Speed of the bullet, v = 100 m/s
The initial kinetic energy of the bullet is given by the equation:
Kinetic energy = (1/2) * mass * velocity^2
Initial kinetic energy of the bullet = (1/2) * 0.01 kg * (100 m/s)^2
= 50 J
Now, we need to find the change in gravitational potential energy of the block. We can calculate this using the equation:
Change in potential energy = mass * gravitational acceleration * change in height
Given:
Mass of the block, M = 990 g = 0.99 kg
b. Find the gravitational potential energy of the block:
Change in potential energy = M * g * change in height
Where:
g = acceleration due to gravity ≈ 9.8 m/s^2
Now, we need the block's change in height. From the conservation of mechanical energy, we know that the initial kinetic energy of the bullet is equal to the change in potential energy of the block.
Initial kinetic energy of bullet = Change in potential energy of block
Substituting the values:
50 J = 0.99 kg * 9.8 m/s^2 * change in height
Now, we can solve for the change in height:
change in height = 50 J / (0.99 kg * 9.8 m/s^2)
≈ 5.1 m
Therefore, the block rises vertically by approximately 5.1 meters.
b. To determine how much of the bullet's energy becomes internal energy, we need to calculate the remaining kinetic energy after the block rises.
Remaining kinetic energy = Initial kinetic energy of the bullet - Change in potential energy of the block
Remaining kinetic energy = 50 J - (0.99 kg * 9.8 m/s^2 * 5.1 m)
Therefore, the remaining kinetic energy represents the amount of energy that became internal energy.
To find the vertical height through which the block rises, we can start by applying the principle of conservation of momentum. The initial momentum of the bullet is given by its mass (10g = 0.01kg) multiplied by its velocity (100m/s). Since the bullet embeds itself in the block, the final momentum of the system is zero because the block and bullet move together after the collision.
Initial momentum = Final momentum
(0.01kg) * (100m/s) = (mass of block + mass of bullet) * final velocity
The mass of the block is 990g, which is equal to 0.99kg.
(0.01kg) * (100m/s) = (0.99kg + 0.01kg) * final velocity
1kg * 100m/s = 1kg * final velocity
final velocity = 100m/s
Now, we can use the principle of conservation of mechanical energy to find the vertical height. When the block rises, it gains potential energy equal to the loss of kinetic energy of the bullet-block system.
The initial kinetic energy of the bullet-block system is given by:
(1/2) * (mass of bullet + mass of block) * (initial velocity)^2
= (1/2) * (0.01kg + 0.99kg) * (100m/s)^2
= 500 J
Since the final velocity is zero, the final kinetic energy of the system is zero.
The loss in kinetic energy is equal to the gain in potential energy, so the potential energy gained by the block is 500 J.
Now, we can use the formula for potential energy:
Potential energy = mass * gravity * height
500 J = (0.99kg) * (9.8m/s^2) * height
Solving for height, we find:
height = 500 J / (0.99kg * 9.8m/s^2)
height ≈ 51.02 meters
Therefore, the vertical height through which the block rises is approximately 51.02 meters.
Now, let's move on to finding how much of the bullet's energy becomes internal energy.
The initial kinetic energy of the bullet-block system is 500 J, as we calculated earlier. This energy transforms into different forms, including internal energy due to the collision between the bullet and the block.
To find the amount of energy that becomes internal energy, we need to subtract the final kinetic energy (0 J) from the initial kinetic energy.
Amount of energy becoming internal energy = Initial kinetic energy - Final kinetic energy
= 500 J - 0 J
= 500 J
Therefore, the bullet's energy transforms into 500 J of internal energy.
Please note that the values might differ depending on rounding and other factors.