Project 10: Part B: Post-Apocalypse Now, Part 2

Stuck in the wilderness for a number of days and unable to contact the people in your bunker, your team along with Willard and Kilgore set out to find a new safe haven. After traversing the scorched wasteland for several days you see postings for “Thunderdome”, which promises “sanctuary for all”. When your team arrives at Thunderdome, you are greeted by the leader of the community Auntie Entity (“Tina” for short), and a 5-story former engineering building with each floor 12 meters high. Your team is told that you all can become community members if you can design a defense system based on a bow design that the community has been working on. The defense system consists of a support anchored to the floor to hold a horizontally aligned spring-loaded harpoon launcher in place. There is a defense system in place on floors 2 through 5, as well as the roof.

In addition to vicious boar tigers, zombies have begun to overrun other settlements in the area and Auntie wants to be prepared for their imminent arrival.

They request some specifics for the machine and indicate some constraints:

• There is an ample supply of 3 kg harpoons.
• The harpoons can only be fired horizontally but must have the greatest variation in range possible.
• It must fire carbon steel harpoons at or below 270 K (it has been found that cooled projectiles have a greater effect on zombies)
• Initial tests suggest that to penetrate zombie flesh, harpoons must have a speed of at least 200 m/s.

The engineering building is 60m from a solid, concrete defensive wall of height 30m and thickness 10m surrounding the Thunderdome. Beyond the wall is a 10m horizontal flat ledge followed by a plain that is 10m below ground level to capture the zombie hoards. Inside the building is an abandoned elevator shaft that extends from the ground floor to the roof. The layout of the engineering building and its surroundings is depicted above.

You are also supplied with the following materials:

• A spring with a lock mechanism that enables it to be locked at various compressions with a spring constant of (15000 N/m). There is no crank strong enough to compress this stiff spring.
• On each floor of the engineering building is one defender, an anchored harpoon launcher support, a harpoon launcher, and one concrete block of mass 400kg resting on an adjustable inclined plane adjacent to the elevator shaft.
• You can also request an amount of ice (at $250\,{\rm K}$) but you have to be specific as supplies are low.

The spring-loaded harpoon launchers are detachable and can easily be moved from floor to floor, or to the bottom of the elevator shaft. The plan is to drop the concrete blocks down the elevator shaft onto the spring to compress the spring. Design the defense system to find the minimum angle for the boulder to fall down the elevator shaft and the range/locations outside the defensive wall that your system can reach with the required specifications. Indicate your supply needs to meet the Aunty Entity's requirements to keep Thunderdome safe.

Overall general idea The general idea is to drop a block into the elevator shaft such that it lands on a vertically aligned harpoon launcher (spring), compressing the spring with the spring latching in place. The launcher can then be brought up to any of the five launcher supports on floors 2 through the roof. The harpoon launcher (spring) is anchored in place here and is horizontally aligned. A harpoon is placed in the launching mechanism where it can be fired at approaching zombies.

The following is broken down into parts for you and your group to use as a (mostly) review problem for next week's second exam.

Part 1: Finding the angle of the platform

We would like to increase the angle of the inclined plane so the block begins to slide on the plane(ramp). The idea here is we want it to fall into the elevator shaft so you can start the next part. Assume the plane is constructed of steel.

1. Do we want to use static or kinetic friction?
2. Draw a free-body diagram of the block on the inclined plane.
3. Find components of each force parallel and perpendicular to the plane.
4. In terms of variables, what are the normal and frictional forces?
5. What is the angle of the plane?

Part 2: Finding the energy stored in the spring The most straightforward distance to find is finding the maximum possible distance a harpoon can reach. This will occur when the block is dropped over the greatest distance (from the roof) and launched from the highest elevation (also from the roof). So, let's work with this first.

1. We want to use the energy principle to find the energy stored in the spring. What are you including in your system?
2. We'd like to use the simplest model, with this in mind what assumptions can we make to simplify this process?
3. What, if anything does work on your system? If any work is done, calculate this work.
4. Apply the energy principle to find the energy stored in your spring. Make your life easier - keep this in variable form!

Part 3: Checking the harpoon speed

1. The minimum speed needed is at the instant the harpoon hits a zombie.
2. In order to find the speed, we need to use energy considerations. What is in your system?
3. What, if anything does work on your system? If any work is done, calculate this work.
4. Keeping your work in variables until the final step (you'll thank us later), apply the energy principle to find the speed.
5. We also will need the launch speed of the arrow so we can find out if it clears the wall. Apply steps 2 through 4 to get this speed, again keep this in variables.

Part 4: Does it clear the wall?

1. Does the time of fall of the horizontally-launched harpoon depend on the launch speed?
2. Is this constant force motion? If so, we can use kinematics here.
3. How much time does it take the harpoon to reach the horizontal location of the wall? Again, keep in variables.
4. What is the elevation of the harpoon at the time found in the previous step - now, use your known quantities to get a numerical value. Does it clear the wall?

Part 5: What is the range/ total horizontal distance of the launched harpoon?

1. Use a similar procedure here as you used in part 4 to find the range with a slight modification.
2. Start with the vertical direction to find the time it takes the harpoon to reach the level of the zombies (hit the zombie).
3. Use the horizontal direction and the time from the previous step to get the range.

Part 6: Amount of ice

1. This is the non-review part of this problem.
2. We'll use the energy principle here. What is in your system?
3. For your given system, what are W and Q?
4. Calculate the amount of ice needed.

Post-Solution questions:

• A 90 kg zombie is moving 1 m/s directly toward Thunderdome. If a harpoon launched with the maximum possible velocity strikes this zombie and embeds in the zombie’s flesh, what is the velocity of the zombie immediately after being struck with the harpoon?
  • The zombie slides with this velocity and friction causes it to come to a stop. What is the work done by friction during this slide? Also, the coefficients of friction between the zombie and the horizontal surface are $\mu_s = 0.3$ and $\mu_k = 0.2$. How far does the zombie slide before coming to a stop?
• Willard suggests you can increase the range of the harpoons if the harpoon launcher is not anchored to the floor, and thereby would recoil. Would this work? Check your prediction by calculation. What is the new launch speed, and what is the recoil speed of the launcher/support? The anchor and spring launcher have a mass of 400 kg (excluding the harpoon).
• Draw a free-body diagram of the harpoon launcher as it slides along the concrete floor. What is the frictional force acting on the launcher during this slide? How far does it slide? After all, you don’t want it to fall into the elevator shaft.
• Sketch the trajectory of one of the launched harpoons. For a point halfway from where it is launched to where it strikes a zombie, draw a free-body diagram showing all forces acting on the harpoon. In the past when objects were in freefall, we used a vertical-horizontal coordinate system. Here, we'd like to investigate what the parallel and perpendicular components of the net force look like, and how they affect the motion. With this in mind, sketch the parallel and perpendicular components of the net force acting on the arrow.

We can use the momentum principle to express the net force as: $$\vec{F}_{net}=\frac{d\vec{p}}{dt}=\frac{d\mid\vec{p}\mid}{dt}\hat{p} + \frac{d\hat{p}}{dt}\mid\vec{p}\mid$$

• Which term is the parallel component of the net force, and which term is the perpendicular component of the net force?
• Which component causes the direction of the momentum to change, and which component causes its magnitude to change?

• Now, find the minimum distance your defending mechanism can reach and still hit a zombie. If you took our suggestions above and kept everything in variables, this is where you thank us.
• What would be the minimum distance of a harpoon if Willard's scheme of not anchoring a launcher were employed?