A 10.0 g bullet is fired into a 1.0 kg block initially at rest on a rough surface and connected to a spring with k = 200N/m. Immediately following the collision the block has a speed of 2.0 m/s. The coefficient of friction between the ground and the block is 0.10.
a) If the ground were frictionless, how far would the block travel before reaching its maximum compression point?
b) The friction between the ground and block cause the block to reach its maximum compression 0.04m before it would on a frictionless surface. What is the mechanical energy of the block-spring system when it has reached this maximum compression point?
c) What is the maximum velocity reached by the block as it returns to its equilibrium point following compress?
To answer these questions, we need to use the principles of momentum, conservation of mechanical energy, and Hooke's law. Let's tackle each part step by step.
a) If the ground were frictionless, we can assume that the only force acting on the block would be the force from the spring. The first step is to calculate the initial velocity of the block immediately after the bullet collides with it.
We can use the law of conservation of momentum, which states that the total momentum before the collision is equal to the total momentum after the collision.
Given:
Mass of bullet (m1) = 10.0 g = 0.01 kg
Mass of block (m2) = 1.0 kg
Initial velocity of the block (v2_initial) = 0 m/s
Final velocity of the block (v2_final) = 2.0 m/s
Using the conservation of momentum equation:
m1 * v1_initial + m2 * v2_initial = m1 * v1_final + m2 * v2_final
Substituting the given values:
0.01 kg * v1_initial + 1.0 kg * 0 m/s = 0.01 kg * 0 m/s + 1.0 kg * 2.0 m/s
Simplifying the equation:
0.01 kg * v1_initial = 2.0 kg * m/s
v1_initial = 2.0 kg * m/s / 0.01 kg
v1_initial = 200 m/s
Now, let's calculate the maximum compression point of the spring. Since there is no friction, the mechanical energy is conserved.
The potential energy stored in the spring (Us) is given by Hooke's law:
Us = (1/2) * k * x^2
To find x (the maximum compression), we need to solve for it.
The mechanical energy of the block-spring system (E) is given by the sum of its kinetic energy (K) and potential energy (Us):
E = K + Us
Since the block is initially at rest (K_initial = 0), we can calculate the mechanical energy at the maximum compression point using only the potential energy:
E = Us = (1/2) * k * x^2
Now, let's move on to part b.
b) The presence of friction between the block and the ground causes the block to reach its maximum compression 0.04 m before it would on a frictionless surface.
To find the mechanical energy of the system at this point, we need to first calculate the amount of work done by friction.
The work done by friction (W_friction) is given by the force of friction multiplied by the distance over which it acts:
W_friction = force_of_friction * distance
The force of friction (f_friction) can be found using the coefficient of friction (μ) and the normal force (N) on the block:
f_friction = μ * N
The normal force (N) is equal to the weight of the block since it is on a horizontal surface:
N = m2 * g
Substituting the given values:
N = 1.0 kg * 9.8 m/s^2
N = 9.8 N
Now we can calculate the force of friction:
f_friction = 0.10 * 9.8 N
f_friction = 0.98 N
Next, we can calculate the work done by friction:
W_friction = 0.98 N * 0.04 m
W_friction = 0.0392 J
Since the mechanical energy (E) is conserved, we can find it using:
E = K + Us
We know the work done by friction is equal to the change in mechanical energy:
W_friction = ΔE = E_final - E_initial
E_initial = 0.5 * k * x^2
E_final = E_initial - W_friction
Now, let's move on to part c.
c) To find the maximum velocity reached by the block as it returns to its equilibrium point following the compression, we can use the principle of conservation of mechanical energy.
The mechanical energy at the maximum compression point (E_final) is known from part b.
Using the principle of conservation of mechanical energy:
E_initial = E_final = K_initial + Us_initial
K_initial = (1/2) * m2 * v2_initial^2
Since the block reaches its maximum compression point and comes to rest before returning to the equilibrium point, the kinetic energy at the maximum compression point (K_final) is zero.
Therefore, we have:
K_initial + Us_initial = 0 + Us_final
Us_final is given by:
Us_final = (1/2) * k * x_max^2
To solve for x_max (maximum displacement), we can rearrange the equation by isolating x_max:
x_max = sqrt((2 * Us_final) / k)
Substituting the given values, we can find x_max.
By solving these equations and plugging in the appropriate values, you can find the answers to all three parts of the question.