Project 14: Part B: Sharks!

Your team is traveling across the Pacific Ocean on a tug boat, and you get a call that someone's boat has broken down, leaving them surrounded by great white sharks. You have a catapult with a steel ball (10 kg) loaded in it, and clay balls (20 kg each). The catapult is 8 m long, has a mass of 5 kg, and rotates about its center. The strongest person on your team is able to throw the clay balls at a speed of 47 m/s. Your mission is to save the people on the stranded boat by hitting the sharks in the snout with the steel ball, and to make sure that the sharks recoil with a great enough speed after being hit by the steel ball that they won't want to come back. Better hurry before it's too late and the sharks attack!

First, we use angular momentum conservation for the collision between the clay ball and the catapult. If we take our system to be the clay ball, the steel ball, and the catapult:

$$L_{i, clay}=L_{f, clay, steel, catapult}$$ $$m_{clay}v_{i,clay}R_{catapult}=I_{tot}w$$ $$m_{clay}v_{i,clay}R_{catapult}=(m_{clay}R_{catapult}^2+m_{steel}R_{catapult}^2+1/12*m_{catapult}L_{catapult}^2)w$$ $$w=7.42 rad/s$$

We can use $v=rw$ to convert this to the linear speed that the steel ball leaves the catapult with:

$$v_{i, steel}=(4 m)*(7.42 rad/s)$$ $$v_{i, steel}=29.68 m/s$$

Now we use energy conservation and linear momentum conservation to consider the collision between the steel ball and the shark. We take our system to be the shark and steel ball, and we assume that the shark starts at rest. We also assume that the collision is elastic. For momentum conservation, we get:

$$p_{i, steel}+p_{i, shark}=p_{f, steel}+p_{f, shark}$$ $$m_{steel}v_{i, steel}=m_{steel}v_{f, steel}+m_{shark}v_{shark}$$

For energy conservation, we get:

$$KE_{i, steel}+KE_{i, shark}=KE_{f, steel}+KE_{f, shark}$$ $$\frac{1}{2}m_{steel}v_{i, steel}^2=\frac{1}{2}m_{steel}v_{f, steel}^2+\frac{1}{2}m_{shark}v_{shark}^2$$

Students will need to look up a mass for a shark. Using 900 kg, solving this system of equations for the final velocities of the steel ball and shark, we get:

$$v_{f, steel}=-29.03 m/s$$ $$v_{shark}=0.65 m/s$$

So, the shark doesn't recoil much, but hopefully enough to scare it away!