A 10 bullet is fired at 200m\s into 1 kg block attached to a spring on a smooth horizontal table if the spring constant is 200N\m. Find the maximum compression of the spring.
To find the maximum compression of the spring, we can use the principle of conservation of mechanical energy. The initial mechanical energy of the system is equal to the sum of the bullet's kinetic energy and the elastic potential energy stored in the spring when it reaches its maximum compression.
First, let's find the initial kinetic energy of the bullet. The formula for kinetic energy is:
Kinetic energy = (1/2) * mass * velocity^2
Since the bullet has a mass of 10 grams (0.01 kg) and a velocity of 200 m/s, we can calculate its initial kinetic energy:
Kinetic energy = (1/2) * 0.01 kg * (200 m/s)^2
= 200 J
Now, let's find the elastic potential energy stored in the spring when it is compressed to its maximum amount. The formula for elastic potential energy is:
Elastic potential energy = (1/2) * spring constant * compression^2
Since the spring constant is given as 200 N/m, we need to find the maximum compression distance that the spring is compressed. To do that, we'll use the equation of motion for an object with constant acceleration:
Final velocity^2 = Initial velocity^2 + 2 * acceleration * displacement
The initial velocity of the bullet is 200 m/s, and the final velocity is 0 m/s since it comes to rest at the maximum compression. The acceleration is given by Newton's second law of motion:
Force = mass * acceleration
In this case, the force on the block is equal to the force exerted by the spring, which is given by Hooke's Law:
Force = spring constant * compression
Setting these two forces equal to each other, we have:
spring constant * compression = mass * acceleration
Rearranging, we get:
compression = (mass * acceleration) / spring constant
Now, let's calculate the acceleration of the block. Since the only force acting on the block is the force exerted by the bullet (from Newton's third law, action and reaction forces are equal and opposite), we can use Newton's second law of motion to find the acceleration:
Force = mass * acceleration
The force exerted by the bullet is equal to the change in momentum, which is given by:
Force = mass * change in velocity / time
Substituting the given values, we have:
Force = 0.01 kg * (200 m/s - 0 m/s) / t
where t is the time it takes for the bullet to come to rest after hitting the block.
Now, let's substitute this expression for acceleration into the earlier equation for compression:
compression = (mass * ((0.01 kg * (200 m/s - 0 m/s) / t))) / spring constant
Finally, the maximum compression of the spring will occur when the bullet comes to rest. So we need to find the time it takes for the bullet to come to rest after hitting the block. We can use the equation of motion:
Final velocity = Initial velocity + acceleration * time
Since the final velocity is 0 m/s and the initial velocity is 200 m/s, we can solve for t:
0 m/s = 200 m/s + (0.01 kg * (200 m/s - 0 m/s)) / t
Solving this equation, we find:
t = 0.01 kg * (200 m/s - 0 m/s) / 200 m/s
= 0.01 s
Now that we have the time, we can calculate the maximum compression of the spring:
compression = (0.01 kg * ((200 m/s - 0 m/s) / 0.01 s)) / 200 N/m
= 0.01 m
Therefore, the maximum compression of the spring is 0.01 meters (or 10 millimeters).