Question

A 17 m long vine hangs vertically from a tree on one side of a 10 m wide gorge. Tarzan runs up, hoping to grab the vine, swing over the gorge, and drop vertically off the vine to land on the other side. At what minimum speed must he be running?

Answer 1

Tarzan needs to run at a minimum speed of approximately 16.39 m/s to swing across a 10 m wide gorge, calculated using the conservation of energy principle and the Pythagorean theorem.

Step-by-step explanation:

To determine the minimum speed Tarzan must be running to swing across a 10 m wide gorge, we can utilize the conservation of energy principle in physics. Assuming Tarzan grabs the vine at the bottom of the swing (the lowest point of the pendulum), his kinetic energy at that point will be completely converted into potential energy at the highest point of the swing on the other side of the gorge. To barely clear the gorge, this highest point will be at the same height as the point where he grabbed the vine.

If we let m be the mass of Tarzan, g the acceleration due to gravity (9.81 m/s2), and h the height he swings up, we can set the kinetic energy at the start (1/2 m v2) equal to the potential energy at the height (mgh). If the gorge is 10 m wide, Tarzan's center of mass must swing up to a height h that forms a right triangle with 10 m as the base and the vine's 17 m length as the hypotenuse.

Using the Pythagorean theorem, we can find h:
h = sqrt(172 - 102) = sqrt(289 - 100) = sqrt(189) = 13.75 m.

Now, equating kinetic and potential energy: 1/2 m v2 = mgh. Canceling mass from both sides and solving for v, we get: v = sqrt(2gh) = sqrt(2 * 9.81 * 13.75) = sqrt(268.725) ≈ 16.39 m/s. Thus, Tarzan needs to run at least 16.39 m/s to swing across the gorge.