# AP Momentum/Energy Problem

Problem
In a laboratory experiment, you wish to determine the initial speed of a dart just after it leaves a dart gun. The dart, of mass m, is fired with the gun very close to a wooden block of mass M0 which hangs from a cord of length l and negligible mass, as shown above. Assume the size of the block is negligible compared to l, and the dart is moving horizontally when it hits the left side of the block at its center and becomes embedded in it. The block swings up to a maximum angle from the vertical. Express your answers to the following in terms of m, M0, l, max, and g.
a. Determine the speed v0 of the dart immediately before it strikes the block.

b. The dart and block subsequently swing as a pendulum. Determine the tension in the cord when it returns to the lowest point of the swing.

c. At your lab table you have only the following additional equipment.

Meter stick Stopwatch Set of known masses
Protractor 5 m of string Five more blocks of mass M0
Spring

Without destroying or disassembling any of this equipment, design another practical method for determining the speed of the dart just after it leaves the gun. Indicate the measurements you would take, and how the speed could be determined from these measurements.

How I tried to solve:

a) Need help.
b) a=v^2/r is centripetal acceleration, multiply by mass for force. And I am now also confused, the problem suggest that the solution should use theta max, L, g, m, Mo. Now if centripetal accerleeration is taken into affect it should be FT = (m+Mo)g - Fc. Because centripetal force is in the same direction as force tension.

You can find Fc by using the maximum y height in potential energy formula, and transforming it into kinetic energy, then find the velocity.
With the velocity you can calculate Fc.
I got 2g(L-Lcos(theta))=v^2
(m+Mo)g - [(m+Mo)(2g(L-Lcos(theta)))]/L = Fnet
Not sure if that is right, I just learned this stuff.
c) Any suggestions would be great, thanks!

I've been trying and I'm still not getting it.
Can someone confirm or reject if b is correct/incorrect please?

Part a) you can solve in terms of energy:

$$K_i = U_{gf}$$

Letter b) seems right, but I'll have to check the actual calculations to be sure. You have the right idea though.

For part c) I would use the stop watch and meter stick to measure the speed of one of the blocks after a collision with the dart. That's just one way though. There are probably others.