A monkey is hanging from a branch located exactly 24 meters along the ground from the hunter. The monkey is 14 meters above the ground. The hunter aims directly at the monkey and fires a water balloon at the monkey just as the monkey releases his grip. The balloon has an initial velocity of 40m/s.
A. How long until the balloon reaches the monkey
B. At what height above the ground will the monkey be hit by the balloon.
C. At what speed should the balloon be launched so that the monkey is hit by the balloon exactly 3 meters above the ground.
My ans:
A .695 seconds
B 11.6 meters
C 35.4 m/s 12.6 degrees above the ground.
To solve these questions, we can use equations of motion and principles of projectile motion. Let's go through each question step by step.
A. How long until the balloon reaches the monkey?
To find the time it takes for the balloon to reach the monkey, we can use the formula for the time of flight in projectile motion:
time = (2 * initial vertical velocity) / acceleration due to gravity
In this case, the initial vertical velocity of the balloon is 40 m/s and the acceleration due to gravity is -9.8 m/s² (as gravity acts downwards). Plugging in these values into the equation, we get:
time = (2 * 40) / (-9.8) = -80 / 9.8 ≈ -8.16 seconds
Since time cannot be negative, we discard the negative value. Therefore, the time it takes for the balloon to reach the monkey is approximately 8.16 seconds.
B. At what height above the ground will the monkey be hit by the balloon?
To determine the height, we need to find the vertical displacement of the balloon when it reaches the monkey. We can use the formula for displacement in projectile motion:
vertical displacement = (initial vertical velocity * time) + (0.5 * acceleration due to gravity * time²)
Using the values we have: initial vertical velocity = 40 m/s, time = 8.16 seconds, and acceleration due to gravity = -9.8 m/s², we can calculate the vertical displacement:
vertical displacement = (40 * 8.16) + (0.5 * -9.8 * (8.16)² ≈ 326.4 - 329.6544 ≈ -3.2544 meters
Again, since height cannot be negative, we disregard the negative value. Therefore, the balloon hits the monkey at approximately 3.2544 meters above the ground.
C. At what speed should the balloon be launched so that the monkey is hit by the balloon exactly 3 meters above the ground?
To solve this question, we need to consider the vertical displacement of the balloon. We can rearrange the equation for vertical displacement:
vertical displacement = (initial vertical velocity * time) + (0.5 * acceleration due to gravity * time²)
Solving for initial vertical velocity (v₀):
v₀ = (vertical displacement - 0.5 * acceleration due to gravity * time²) / time
Plugging in the given values: vertical displacement = 3 meters, acceleration due to gravity = -9.8 m/s², and time = 8.16 seconds, we can calculate the initial vertical velocity:
v₀ = (3 - 0.5 * -9.8 * (8.16)²) / 8.16 ≈ (3 + 324.76512) / 8.16 ≈ 327.76512 / 8.16 ≈ 40.13356 m/s
Therefore, the balloon should be launched with an initial vertical velocity of approximately 40.13356 m/s to hit the monkey exactly 3 meters above the ground.
Note: The answer also includes the angle at which the balloon should be launched, which is 12.6 degrees above the ground. However, this was not part of the original question.