A 10g bullet is fired at 200m/s into a 1kg 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 need to understand the concept of conservation of momentum and conservation of mechanical energy.
First, let's find the initial momentum of the bullet. The momentum (p) is given by the product of mass (m) and velocity (v):
Initial momentum (p1) = mass of bullet (m1) × velocity of bullet (v1)
= 10g × 200m/s
= 0.01 kg × 200 m/s
= 2 kg m/s
Next, we need to find the final momentum of the bullet and the block after the collision. Since the bullet is embedded in the block and they move together, we can consider it as a single system. The final momentum of the system after the collision (p2) will be zero since there is no external force acting in the horizontal direction.
Now, let's apply the principle of conservation of momentum:
Initial momentum (p1) = Final momentum (p2)
2 kg m/s = 0
From this equation, we can see that the final velocity of the bullet and block system will be zero after the collision.
Next, let's consider the conservation of mechanical energy. Before the collision, the bullet had kinetic energy, and after the collision, this energy will be converted into potential energy stored in the compressed spring.
The kinetic energy (KE) of the bullet before the collision can be calculated using the formula:
KE = 1/2 × mass × velocity^2
= 1/2 × 0.01kg × (200m/s)^2
= 200 J
Since the final velocity is zero, the kinetic energy after the collision will be zero.
Therefore, the change in mechanical energy (ΔE) is equal to the kinetic energy before the collision:
ΔE = KE = 200 J
Now, let's use the formula for the potential energy stored in a spring:
Potential energy (PE) = 1/2 × spring constant (k) × compression^2
The maximum compression of the spring occurs when all the initial kinetic energy of the bullet is converted into potential energy in the spring. Therefore, we can equate the change in mechanical energy to the potential energy stored in the spring:
200 J = 1/2 × 200 N/m × compression^2
Simplifying this equation, we find:
compression^2 = 200 J / (100 N/m) = 2 m
Taking the square root of both sides, we get:
compression = √2 ≈ 1.41 m
Therefore, the maximum compression of the spring is approximately 1.41 meters.