Final answer:
To determine the velocity with which the banana must be launched to hit the free-falling monkey, one must apply the principles of projectile motion and free fall. It requires the horizontal travel time of the banana to match the monkey's fall time due to gravity. The calculated velocity is slightly above 22 m/s, but option b) 18 m/s is the closest given answer.
Step-by-step explanation:
To solve the problem of the banana hitting a monkey that falls from the tree the moment the banana is launched, we have to use the principles of projectile motion and free fall.
Given that the monkey is 100 meters above the ground and the banana is launched horizontally at the same height, the only force acting on both the monkey and the banana (in the vertical direction) is gravity.
Since gravity affects all objects equally regardless of their horizontal motion, the banana will only hit the monkey if it maintains the same vertical position relative to the monkey while traveling horizontally.
For the banana to hit the monkey, the time it takes for the banana to travel horizontally 100 meters must be equal to the time it takes for the monkey to fall 100 meters due to gravity.
The time of fall (t) for the monkey can be calculated using the formula for free fall distance d = 1/2 g t^2, where g is the acceleration due to gravity (9.8 m/s2). Solving for t gives us t = sqrt(2d/g). Plugging in the numbers t = sqrt(2*100m/9.8 m/s2), the time comes out to approximately 4.52 seconds.
To find the required horizontal velocity (v) of the banana, we use the formula v = d/t. Thus, v = 100m / 4.52s, which gives us a velocity slightly above 22 m/s.
However, since this value isn't one of the options, we may consider that the question assumes a rounded value of gravitational acceleration to be 10 m/s2 for simplicity, thus making the calculation easier and matching one of the given options more closely.
With g = 10 m/s2, t would be exactly 4.47 seconds, and v would then be approximately 22.37 m/s. Therefore, the closest provided answer to this calculated velocity would be option b) 18 m/s, but none of the options correspond exactly to the calculated velocity.