A 17mlong vine hangs vertically from a tree on one side of a 10 m wide gorge, as shown in the figure. 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.
I do 17^2+10^2=c^2 and get 19.7
then I subtract 19.7 from 17 and get 2.7 then I use the formula v=sqr(2gh
and for g I use 9.8 and for h I get 2.7
my final answer is 7.7, where did i go wrong?
The pendulum swings out an angle T
sin T = 10/17 = .588
so T = 36 degrees
how high?
17 - 17 cos T = 3.25 high
Potential energy at 3.25 = m gh = m(9.8)(3.25) = 31.85 * m Joules
so
(1/2) m v^2 = 31.85 m
v^2 = 63.7
so v is about 8 m/s
Let's go through the steps again to find where the mistake might have occurred:
Step 1: Find the length of the vine using the Pythagorean theorem.
17^2 + 10^2 = c^2
289 + 100 = c^2
389 = c^2
c ≈ 19.7 m (this step is correct)
Step 2: Determine the difference in height between Tarzan's starting position and the lowest point of the vine.
The difference in height is the height of the tree minus the length of the vine.
h = 17 - 10 = 7 m (this step is correct)
Step 3: Calculate the minimum speed necessary for Tarzan to clear the gorge.
Using the conservation of energy principle and the equation v = sqrt(2gh):
v = sqrt(2 * 9.8 * 7)
v = sqrt(137.2)
v ≈ 11.7 m/s (this step is correct)
Therefore, the correct answer should be approximately 11.7 m/s as the minimum speed Tarzan must be running.
To determine the minimum speed required for Tarzan to successfully swing across the gorge, we can break down the problem into different parts.
1. Finding the distance Tarzan needs to swing:
Using the Pythagorean theorem, you correctly found the hypotenuse of the right triangle formed by the vine, which is √(17^2 + 10^2) = 19.7 meters. However, Tarzan needs to swing over the gorge, so the distance he needs to cover is twice the gorge width, which is 2 * 10 = 20 meters.
2. Calculating the height Tarzan needs to reach:
Since Tarzan drops vertically off the vine to land on the other side, the height he needs to reach is the same as the width of the gorge, which is 10 meters.
3. Applying the equation of motion to find Tarzan's minimum speed:
The equation you mentioned, v = √(2gh), is correct. However, you miscalculated the values of g and h. The value of g, which represents the acceleration due to gravity, is approximately 9.8 m/s^2, as you correctly used. But the height Tarzan needs to reach (h) is not 2.7 meters. It is equal to the width of the gorge, which is 10 meters.
Using the correct values, the equation becomes:
v = √(2 * 9.8 * 10) = √(196) = 14 m/s
Therefore, Tarzan needs to be running with a minimum speed of 14 m/s to successfully swing across the gorge.