A bullet of mass 1.9×10−3 kg embeds itself in a wooden block with mass 0.987 kg, which then compresses a spring (k = 120 N/m) by a distance 3.5×10−2 m before coming to rest. The coefficient of kinetic friction between the block and table is 0.42. What fraction of the bullet's initial kinetic energy is dissipated (in damage to the wooden block, rising temperature, etc.) in the collision between the bullet and the block?
(delta K) / K = ?
To find the fraction of the bullet's initial kinetic energy dissipated in the collision, we need to determine the initial kinetic energy of the bullet and the final kinetic energy of the bullet-block system.
1. Calculate the initial kinetic energy (K_i) of the bullet:
K_i = (1/2) * m_bullet * v_bullet^2
Given:
m_bullet = 1.9×10^(-3) kg
v_bullet = velocity of the bullet (not provided)
2. Calculate the final kinetic energy (K_f) of the bullet-block system:
K_f = (1/2) * (m_bullet + m_block) * v_final^2
Given:
m_block = 0.987 kg
v_final = velocity of the bullet-block system after collision (not provided)
3. Calculate the work done by the friction force (W_friction) during the compression of the spring:
W_friction = f_friction * d
Given:
f_friction = coefficient of kinetic friction * normal force
d = distance compressed by the spring = 3.5×10^(-2) m
4. Use the work-energy theorem to find the change in kinetic energy (ΔK) of the bullet-block system:
ΔK = K_f - K_i = -W_friction
5. Calculate the fraction of the bullet's initial kinetic energy dissipated:
(delta K) / K = (ΔK) / K_i
Note: To fully solve this problem, we need additional information about the velocity of the bullet and the velocity of the bullet-block system after the collision.
To determine the fraction of the bullet's initial kinetic energy that is dissipated, we need to calculate the change in kinetic energy (ΔK) of the bullet during the collision and compare it to the bullet's initial kinetic energy (K).
The initial kinetic energy (K) of the bullet can be calculated using the formula:
K = (1/2) * mass * velocity^2
First, we need to calculate the velocity of the bullet. We know the bullet's mass (m) is 1.9×10^(-3) kg. However, we need more information to determine the bullet's velocity.
We are given that the bullet embeds itself in the wooden block, which means both the bullet and block move together after the collision. To find the velocity of the bullet-block system, we can use the law of conservation of momentum:
Initial momentum = Final momentum
m1 * v1 = (m1 + m2) * vf
where m1 is the mass of the bullet, m2 is the mass of the block, v1 is the initial velocity of the bullet, and vf is the final velocity of the bullet-block system.
In this case, the final velocity (vf) is zero because the bullet and block come to rest. Substituting the known values, we have:
(1.9×10^(-3) kg) * v1 = (1.9×10^(-3) kg + 0.987 kg) * 0
Simplifying the equation, we find that v1 = 0 m/s.
Since the initial velocity of the bullet is zero, its initial kinetic energy (K) is also zero.
Now let's calculate the change in kinetic energy (ΔK) of the bullet during the collision. The change in kinetic energy can be calculated using the formula:
ΔK = Kf - Ki
where Kf is the final kinetic energy of the bullet after the collision.
Since the bullet comes to rest after embedding itself in the wooden block, the final velocity of the bullet-block system is zero.
Thus, the final kinetic energy (Kf) is also zero.
Substituting the values into the equation, we have:
ΔK = 0 - 0
Hence, the change in kinetic energy (ΔK) of the bullet during the collision is zero.
To calculate the fraction of the bullet's initial kinetic energy dissipated, we will divide the change in kinetic energy (ΔK) by the initial kinetic energy (K):
(delta K) / K = ΔK / K = 0 / 0
Since any number divided by zero is undefined, we cannot determine the fraction of the bullet's initial kinetic energy dissipated in this collision.
(1) Use the compression of the spring to deduce the velocity of the block with embedded bullet, immediately after impact. Assume all of the kinetic energy of the block/bullet combination is converted to spring potential energy PLUS frictional work as the block drags over the table.
(2) Divide the initial KE of the bullet by the combined bullet/block KE that you get from part 1.
Show your work if you wish further assistance