The collision between a hammer and a nail can be considered to be approximately elastic. Estimate the kinetic energy acquired by a 13 g nail when it is struck by a 550 g hammer moving with an initial speed of 4.0 m/s.

I solved for the final velocities of both and got 3.8 m/s for the nail and 8.46 m/s for the nail.

How do I get my final answer.. I'm so close! haha

3.8 m/s for the hammer*

Assuming that you have the correct velocity for the nail after it is hit (8.46 m/s), call that V and compute its kinetic energy with the usual formula,

KE = (1/2) M V^2

They only want the K.E. of the nail.

To determine the kinetic energy acquired by the nail after it is struck by the hammer, you can use the equation for kinetic energy:

Kinetic energy = (1/2) * mass * velocity^2

Let's calculate the kinetic energy acquired by the nail:

Given:
Mass of the nail (m_nail) = 13 g = 0.013 kg
Mass of the hammer (m_hammer) = 550 g = 0.55 kg
Initial velocity of the hammer (v_initial_hammer) = 4.0 m/s
Final velocity of the nail (v_final_nail) = 3.8 m/s
Final velocity of the hammer (v_final_hammer) = 8.46 m/s

First, we need to calculate the final velocity of the hammer-nail system (v_final_system). This can be calculated using the principle of conservation of momentum. The momentum before the collision (p_initial) is equal to the momentum after the collision (p_final):

p_initial = p_final

(m_hammer * v_initial_hammer) = (m_hammer * v_final_hammer) + (m_nail * v_final_nail)

Substituting the given values:

(0.55 kg * 4.0 m/s) = (0.55 kg * v_final_hammer) + (0.013 kg * 3.8 m/s)

Solving this equation will give us the final velocity of the hammer-nail system (v_final_system).

Once we have the final velocity of the system, we can calculate the kinetic energy acquired by the nail using its final velocity (v_final_nail):

Kinetic energy = (1/2) * m_nail * v_final_nail^2

Substituting the given values, including the final velocity of the nail (v_final_nail) that you have already calculated, will give you the final answer.

To find the kinetic energy acquired by the nail, you need to calculate the initial and final kinetic energies and then find the difference between them.

The formula for kinetic energy is KE = 1/2 * m * v^2, where KE is the kinetic energy, m is the mass, and v is the velocity.

First, calculate the initial kinetic energy of the hammer using its mass (550 g) and initial velocity (4.0 m/s):
KE_hammer_initial = 1/2 * 0.550 kg * (4.0 m/s)^2

Next, calculate the final kinetic energy of the hammer using its mass (550 g) and final velocity (8.46 m/s):
KE_hammer_final = 1/2 * 0.550 kg * (8.46 m/s)^2

For the nail, do the same calculation. Calculate the initial kinetic energy using its mass (13 g) and initial velocity (0 m/s) since it was initially at rest:
KE_nail_initial = 1/2 * 0.013 kg * (0 m/s)^2

Then, calculate the final kinetic energy using its mass (13 g) and final velocity (3.8 m/s):
KE_nail_final = 1/2 * 0.013 kg * (3.8 m/s)^2

Finally, find the difference between the final and initial kinetic energies of the nail:
Delta_KE_nail = KE_nail_final - KE_nail_initial

The value of Delta_KE_nail is the kinetic energy acquired by the nail when it is struck by the hammer.