course_planning:183_projects:f17_project_15_solution [Projects & Practices in Physics]

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====== 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!


<WRAP tip>
==Tutor Questions:==

  * **Question:**  What did you choose for your system?
  * **Expected Answer:**  For the clay ball-catapult collision, we included the clay ball, the steel ball, and the catapult. For the steel ball-shark collision, we included the steel ball and the shark.

  * **Question:**  What was conserved in each collision?
  * **Expected Answer:**  In the first collision, only angular momentum was conserved. Angular momentum is conserved because there were no external torques. The pivot point exerts an external force on the system, so linear momentum is not conserved. It is an inelastic collision, so energy is not conserved. In the second collision, linear momentum is conserved because there are no external forces, and energy is conserved because we assume the collision is elastic.


  * **Question:**  What assumptions did you make?
  * **Expected Answer:**  The water exerts no force resisting the shark's recoil. That gravity doesn't accelerate the steel ball its way from the catapult to the shark. That the steel ball experiences no air resistance. That the clay ball hits the catapult perpendicular to its arm at the very end of the arm. That the shark-steel ball collision is elastic.
</WRAP>