A 250g bullet with a speed of 500m/s passes through the center of a 1.5kg square wooden block [ side length 18cm] at rest upon a horizontal frictionless surface. As it exits the block, the bullet has a speed of 150m/s.
A) find the initial speed of the center of mass of the system
b) Find the final speed of the wooden block
C) what fraction of the bullets initial kinetic energy is lost?
D) where does lost energy go?
E) how far does the block move while the bullet is passing throught it?
A) To find the initial speed of the center of mass of the system, we can use the principle of conservation of momentum. The initial momentum of the system is equal to the final momentum of the system. The momentum of an object is given by the product of its mass and velocity.
The initial momentum is the sum of the momentum of the bullet and the block before the collision. The momentum of the bullet is given by its mass (250g = 0.25kg) multiplied by its initial speed (500m/s). The momentum of the block is zero since it is at rest.
So the initial momentum of the system is 0.25kg * 500m/s = 125kg*m/s.
The final momentum of the system is the sum of the momentum of the bullet and the block after the collision. The momentum of the bullet is given by its mass (0.25kg) multiplied by its final speed (150m/s). The momentum of the block is given by its mass (1.5kg) multiplied by its final speed (which we need to find).
Using the principle of conservation of momentum, we can set the initial momentum equal to the final momentum:
125kg*m/s = (0.25kg * 150m/s) + (1.5kg * final speed)
Simplifying the equation, we get:
125kg*m/s = 37.5kg*m/s + (1.5kg * final speed)
Subtracting 37.5kg*m/s from both sides, we get:
125kg*m/s - 37.5kg*m/s = 1.5kg * final speed
87.5kg*m/s = 1.5kg * final speed
Finally, we can solve for the final speed of the wooden block:
final speed = 87.5kg*m/s / 1.5kg = 58.33m/s
Therefore, the initial speed of the center of mass of the system is 58.33m/s.
B) We have already found the final speed of the wooden block in part A, which is 58.33m/s.
C) To find the fraction of the bullet's initial kinetic energy that is lost, we need to first find the initial kinetic energy of the system (bullet + block) before the collision and the final kinetic energy of the bullet after the collision.
The initial kinetic energy of the system is given by:
Initial KE = 0.5 * mass * (initial speed)^2 = 0.5 * (0.25kg + 1.5kg) * (500m/s)^2
The final kinetic energy of the bullet is given by:
Final KE = 0.5 * mass * (final speed)^2 = 0.5 * (0.25kg) * (150m/s)^2
Now we can calculate the fraction of the initial kinetic energy that is lost:
Fraction lost = (Initial KE - Final KE) / Initial KE
Calculate the values and find the fraction of the initial kinetic energy that is lost.
D) The lost energy goes into other forms, such as heat generated by the collision, sound, and possibly the deformation of the wooden block and the bullet.
E) To determine how far the block moves while the bullet is passing through it, we need to analyze the conservation of linear momentum.
The initial momentum of the system is given by the sum of the momenta of the bullet and the block. The momentum of the bullet is equal to its mass multiplied by its initial speed, which we know to be 0.25kg * 500m/s = 125kg*m/s. The momentum of the block is zero since it is at rest.
As the bullet exits the block, the total momentum of the system remains constant. The momentum of the bullet is given by its mass (0.25kg) multiplied by its final speed (150m/s). The momentum of the block is given by its mass (1.5kg) multiplied by its final speed (which we need to find).
Setting the initial momentum equal to the final momentum gives us:
125kg*m/s = (0.25kg * 150m/s) + (1.5kg * final speed)
Simplifying the equation, we get:
125kg*m/s = 37.5kg*m/s + (1.5kg * final speed)
Subtracting 37.5kg*m/s from both sides:
125kg*m/s - 37.5kg*m/s = 1.5kg * final speed
87.5kg*m/s = 1.5kg * final speed
Finally, solving for the final speed of the block:
final speed = 87.5kg*m/s / 1.5kg = 58.33m/s
Since the block is initially at rest, the distance it moves is given by the equation:
distance = (1/2) * (final speed)^2 / acceleration
In this case, the acceleration is zero because the block is on a horizontal frictionless surface. Thus, the block will not move while the bullet is passing through it.