Question

Indiana Jones attempts t0 cross a river by swinging on a vine that is 10 m long: To do this, he pushes off with a speed of 5 mls: The vine initially makes an angle of 300 relative to the vertical. a) What is his speed at the bottom of the swing? b) If he fails to jump off on the other side, how high will he rise on the other side relative to the bottom of the swing?

Answer 1

a) Indiana Jones' speed at the bottom of the swing is 14 m/s. b) Indiana Jones will rise to a height of 10.2 meters on the other side.

Step-by-step explanation:

Indiana Jones swinging on a vine

a) Speed at the bottom of the swing
To determine Indiana Jones' speed at the bottom of the swing, we need to use the principle of conservation of mechanical energy. At the top of the swing, all of the potential energy is converted to kinetic energy. Therefore, we can equate the initial potential energy to the final kinetic energy:
mgh = 0.5mv^2
Where m is the mass, g is the acceleration due to gravity, h is the height, and v is the velocity.
Given that the vine makes an angle of 30 degrees with the vertical, we can calculate the height as h = 10sin(30) = 5 meters.
Using the equation above and substituting the given values, we get:
(5)(9.8) = 0.5m(v^2)
Simplifying, we find:
(49) = 0.5(0.5)(v^2)
(49) = 0.25(v^2)
(v^2) = 49/0.25
(v^2) = 196
Taking the square root of both sides, we find:
v = 14 m/s
Therefore, Indiana Jones' speed at the bottom of the swing is 14 m/s.

b) Height reached on the other side
To determine the height Indiana Jones will reach on the other side, we can use the conservation of mechanical energy as well. At the bottom of the swing, all of the kinetic energy is converted to potential energy. Therefore, we can equate the initial kinetic energy to the final potential energy:
0.5mv^2 = mgh
Using the same values of mass, g, and v as before, and solving for h, we find:
0.5(0.5)(14^2) = h(9.8)
h = (0.5(0.5)(14^2))/(9.8)
Simplifying, we get:
h = 10.2 m
Therefore, Indiana Jones will rise to a height of 10.2 meters relative to the bottom of the swing on the other side.

Answer 2

Indiana Jones' problem of crossing a river by swinging on a vine is a physics question that deals with the conservation of mechanical energy, involving calculations for speed at the bottom of the swing and the height he reaches on the other side if he fails to jump off.

Step-by-step explanation:

To answer part a), Indiana's speed at the bottom of the swing can be determined by considering the conservation of mechanical energy, which states that the sum of kinetic and potential energy in a system remains constant if no external work is done on the system.

At the highest point of the swing, with the vine at a 30-degree angle to the vertical, the potential energy is at its maximum and the kinetic energy is partly determined by his initial push-off speed. As Indiana swings down to the bottom of the arc, his potential energy converts into kinetic energy, which means his speed increases. At the bottom, where his potential energy is at a minimum, his kinetic energy and, thus, speed will be at a maximum. This speed can be found using the equation for kinetic energy ½ mv^2 and the equation for potential energy mgh, keeping in mind that the total mechanical energy at the top equals the total mechanical energy at the bottom.

For part b), if Indiana fails to jump off on the other side, he will rise until his kinetic energy is completely converted into potential energy again. The height he will reach can be determined using the same principles of energy conservation. Since there are no external forces doing work (ignoring air resistance and the vine's mass), he will rise to a height that corresponds to the amount of kinetic energy he had at the bottom of the swing.