(1) 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 = ?
(2) A neutron collides elastically with a helium nucleus (at rest initially) whose mass is four times that of the neutron. The helium nucleus is observed to move off at an angle of 45 degrees. The neutron's initial speed is 5.9×105 m/s. Determine the speeds of the two particles after the collision.
To solve these problems, we will use the principles of conservation of momentum and conservation of kinetic energy.
For problem (1), to find the fraction of the bullet's initial kinetic energy that is dissipated, we need to calculate the initial and final kinetic energies of the system.
Step 1: Calculate the initial total momentum:
The initial momentum is conserved, so we can write:
p_initial = p_final
The initial momentum is given by:
p_initial = m_bullet * v_bullet
Step 2: Calculate the final total momentum:
The bullet embeds itself in the wooden block, so the combined mass of the bullet and the block moves together after the collision. Let's call this mass "m_combined."
The final momentum is given by:
p_final = m_combined * v_final
Step 3: Calculate the change in kinetic energy and the initial kinetic energy:
The change in kinetic energy (delta K) is given by:
delta K = K_initial - K_final
The initial kinetic energy is given by:
K_initial = (1/2) * m_bullet * v_bullet^2
Step 4: Calculate the dissipated energy fraction:
Finally, the fraction of the bullet's initial kinetic energy that is dissipated is given by:
(delta K) / K_initial
Now, let's go through the steps again to solve problem (2):
Step 1: Calculate the initial total momentum:
The initial momentum is conserved, so we can write:
p_initial = p_final
The initial momentum is given by:
p_initial = m_neutron * v_neutron
Step 2: Analyze the collision:
Since the collision is elastic, both momentum and kinetic energy are conserved.
Step 3: Determine the final velocities:
We have two unknowns: the final velocity of the neutron (v_neutron_f) and the final velocity of the helium nucleus (v_helium_f). We can use the fact that the helium nucleus moves off at an angle of 45 degrees to solve for these velocities.
Step 4: Apply conservation of momentum and solve for the final velocities:
Using the principles of conservation of momentum, we can write two equations:
m_neutron * v_neutron = m_neutron * v_neutron_f + m_helium * v_helium_f (equation 1)
m_neutron * v_neutron^2 = m_neutron * v_neutron_f^2 + m_helium * v_helium_f^2 (equation 2)
Step 5: Solve the equations for the final velocities:
You can solve equations 1 and 2 simultaneously to find the values of v_neutron_f and v_helium_f.
By following these steps, you should be able to solve both problems and find the answers you're looking for.