A monkey swings on a vine and at the bottom of the swing while moving horizontally lets go of the vine aiming for another vine 19 meters away horizontally and whose bottom is 13.5 meters below the monkey. How fast must the monkey be moving to the nearest tenth of a m/s in order that he just catches the bottom of the vine?
To find the speed at which the monkey must be moving to catch the bottom of the vine, we can use the principle of conservation of energy. The energy at the highest point of the swing is equal to the energy at the lowest point, neglecting air resistance.
Let's break down the problem into horizontal and vertical components.
1. Horizontal component:
The distance between the two vines is 19 meters, which represents the horizontal displacement. As the monkey swings, there is no horizontal force acting on it, so its horizontal velocity remains constant throughout the swing. Therefore, the speed needed to reach the other vine horizontally is the same as the horizontal speed at the bottom of the swing.
2. Vertical component:
The vertical displacement between the two vines is 13.5 meters. The potential energy lost by the monkey at the top of the swing is converted into kinetic energy when it reaches the bottom point. The kinetic energy is given by the formula: KE = 1/2 * m * v^2, where m is the mass of the monkey and v is its velocity.
The potential energy at the top of the swing is equal to the mass times the gravitational acceleration times the height (PE = m * g * h). Since the mass and gravitational acceleration are constant, we can ignore them for the purpose of finding the relationship between potential and kinetic energy.
PE = KE
m * g * h = 1/2 * m * v^2
g * h = 1/2 * v^2
v^2 = 2 * g * h
v = sqrt(2 * g * h)
Now, we can solve for v by substituting the values of g (acceleration due to gravity) and h (vertical displacement) into the equation. The acceleration due to gravity is approximately 9.8 m/s^2.
v = sqrt(2 * 9.8 * 13.5)
v ≈ 14.0 m/s (rounded to the nearest tenth of a m/s).
Therefore, the monkey must be moving at a speed of approximately 14.0 m/s horizontally to catch the bottom of the vine.