A particle of mass m moving at 5 m/s in the positive x direction makes a glancing elastic collision with a particle of mass 2m that is at rest before the collision. After the collision, m moves off at an angle of 45 degrees to the x axis and 2m moves off at 60 degrees to the x axis. What is the speed of m after the collision?
To find the speed of particle "m" after the collision, we can use the principle of conservation of momentum and the principle of conservation of kinetic energy.
First, let's find the initial momentum of the system. The initial momentum is given by the sum of the individual momenta of the particles before the collision. Since the second particle is at rest, its initial momentum is zero. The initial momentum of particle "m" is given by:
P_initial_m = m * v_initial_m
where m is the mass of particle "m", and v_initial_m is its initial speed. Given that v_initial_m = 5 m/s, we have:
P_initial_m = m * 5
Now, let's find the final momentum of the system. The final momentum is the sum of the individual momenta of the particles after the collision. The momentum of particle "m" after the collision can be broken down into its x and y components. The x-component of its momentum is given by:
P_final_m_x = m * v_final_m * cos(45°)
where v_final_m is the speed of particle "m" after the collision. Similarly, the y-component of its momentum is given by:
P_final_m_y = m * v_final_m * sin(45°)
For the other particle of mass 2m, the x-component of its momentum after the collision is given by:
P_final_2m_x = 2m * v_final_2m * cos(60°)
where v_final_2m is the speed of the 2m particle after the collision. Similarly, the y-component of its momentum is given by:
P_final_2m_y = 2m * v_final_2m * sin(60°)
Using the conservation of momentum principle, the sum of the initial momenta is equal to the sum of the final momenta:
P_initial_m = P_final_m_x + P_final_2m_x
P_final_m_y = P_final_m_y + P_final_2m_y
Now, let's use the conservation of kinetic energy principle. The kinetic energy before the collision is given by:
KE_initial = 1/2 * m * v_initial_m^2
The kinetic energy after the collision is given by:
KE_final = 1/2 * m * v_final_m^2 + 1/2 * 2m * v_final_2m^2
Using the conservation of kinetic energy principle, the initial kinetic energy is equal to the final kinetic energy:
KE_initial = KE_final
Now, we can substitute the expressions for the momenta and simplify the equations to solve for the unknowns.
P_initial_m = P_final_m_x + P_final_2m_x
P_final_m_y = P_final_m_y + P_final_2m_y
KE_initial = KE_final
By substituting the given values for the angles (45° and 60°), you can solve these equations to find the values of v_final_m and v_final_2m.