P2F10-Tut7-soln - Week 7 Tutorial 1. Homer Simpson is...

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Unformatted text preview: Week 7 Tutorial 1. Homer Simpson is trying to jump the Grand Canyon so he builds a ramp at a 35 degree angle. Assume that the canyon is the same height on both sides, the width of the canyon is 625 yards (1 ft = .3048m), and Homer has an initial speed of 70m/s. a. How long does it take for Homer to return to his take off height? b. What is Homer’s maximum height? c. Does Homer make it across the Grand Canyon in triumph or end up in the hospital? 2. Wile E Coyote is on top of a 100m high cliff and is trying to capture the Road Runner below. He decided to purchase a catapult from ACME, but the catapult can only fire horizontally and the release point is 5 meters above the cliff. At what velocity should Wile E fire the catapult if he wants to hit the Road Runner whom is 200m away from the base of the cliff? 3. Robin Hood fires an arrow at 16m/s at 10 degrees above the horizontal and hits a target located 6 m below his release point. a. What was the flight time of the arrow? b. How far away was the target in the x direction from Robin Hood? c. Where was the arrow located and which direction is the arrow moving when half of the flight time elapsed? 4. Captain Hook is trying to hit Peter Pan with a cannon. Pan is located on the shore that is 30 m away from the cannon. If the cannon fires at 25m/s, at what angle should Hook aim to hit Peter Pan. [Hint, use the fact that ] 5. Tim Howard kicks a soccer ball 80 yards down the field in 4 seconds. If the ball was kicked at a 60 degree angle, what is the position vector of the soccer ball after 3 seconds? 6. If a particle has an initial velocity of Vo = (4.5m/s, 6m/s) and has an acceleration with magnitude 3.0m/s^2 at an angle 20 degrees north of west, what is the position vector and velocity vector at time 2.2 seconds? 1. a. We can find the time it takes for Homer to return back to the take off height by setting the final y position equal to the initial y position and then solving for t. b. We can find Homer’s max height by using the fact that the velocity in the y direction at max height is 0 and plug in our known values. c. To determine if Homer makes the jump, we need to look at the distance Homer travels in the x direction when Homer returns to his original take off height in the y direction. We do that since the canyon is the same height on both sides. Since 625 yards converts to 571.5m, and Homer only went 470m in the x direction by the time he returned to the takeoff height, He didn’t make the jump. 2. Using the fact that the initial velocity in the y direction is 0 since the catapult is fired horizontally, we can solve for the time it takes for the projectile to hit the ground. Then we use that time along with how far the projectile must go in that time to figure out the initial velocity. 3. a. To find the flight time of the arrow, we set the displacement in the y direction to be -6m and then solve for t using all of the other known values. b. To find how far away the target is in the x direction, we use the fact that the target is located at the same x position as the arrow at the time found in part a since the arrow hits the target then. c. The location of the arrow is described by the position vector. Use the fact that half of the flight time is just half of the time found in part a and then plug in the other known values to find your x position and your y position. To find the direction the arrow is moving, we must look at the velocity vector. To find this, we can either take the derivative of the position vector with respect to time, or just use kinematic equations again. 4. To figure out the launch angle needed, we use the fact that the time when the projectile returns to its initial launch height (y direction), the projectile must also reach the shore (x direction). We can then solve for time in the y direction equation in terms of , , and g. Plugging that time into the x direction equation, and then plugging in our known values, we get the angle needed. 5. To find the position vector of the soccer ball at t=3, we must first find the initial velocity of the soccer ball. First we must convert yards to meters which returns 73.152m for x at t=4. Then we can use that information along with our other knowns to find the initial velocity. Then we use that initial velocity and kinematic equations to find our position vector like we did in 3c. 6. In this problem, the acceleration is no longer g, but instead the acceleration vector given in the problem. The acceleration is given in polar form so we must convert that into Cartesian(x,y) format. The initial velocity vector is already given in Cartesian so nothing needs to be changed with that. From there, we use the same approach in 3c. ...
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