A paintball of mass 30 g is fired horizontally at a pendulum which is at rest.

Question

A paintball of mass 30 g is fired horizontally at a pendulum which is at rest. The mass of the pendulum is 60 g and after being hit by the paintball, which sticks to the pendumum, it swings to a height of 20 cm.

What was the intitial speed of the paintball?

My professor gave us the basic stuff about inelastic collisions, I cannot use my notes to refer to so if someone can help me I would really appreciate it. WE HAVE AN EXAM NEXT WEEK!

Answer 1

Consider the mass on the end of the pendulum after it has swung up to its height of 20 cm. If you take a snapshot at that point when it is instantaneously stopped (just prior to it swinging backwards), what is the energy in the system? The mass is not moving, so it has zero velocity and consequently zero kinetic energy. The energy in the system is comprised wholly on gravitational potential energy which is calculated as:

PE = mgh

The mass is 60g + 30g = 90g = 0.09 kg

The height is 20 cm = 0.20 m

So PE = (0.09)(0.2)(9.81) = 0.17658 joules

We will assume no energy is lost during the process of the paintball hitting the pendulum and sticking causing the pendulum to move.

So, because energy is conserved, the energy the system had initially (just after the paintball took off for the pendulum) is the same as the energy after it struck the pendulum and that energy is the same as the energy the system has when the pendulum is at the top of its swing.

So the initial energy (E_i) equals the final energy (E_f)

E_i = the kinetic energy of the paintball = 1/2mv² = 1/2(0.03)v² = 0.015v²

E_f = the potential energy at the top of the pendulum's swing = mgh = 0.17658 joules

So since E_i = E_f, then 0.17658 = 0.015v² or v² = 11.772 and v = √11.772 = 3.4 m/s