Solve for Coconut Drop Height: Kinematics Help on a Sunny Beach Day

[bukky]

Hi,
I just joined today and would like help with a question. Here it is:

It is a sunny day and you are lying on a beach . Some distance away across level gropund, is a tall coconut palm. As you watch, a monkey in that tree drops a coconut(from rest) onto hard ground below. The time elapsed from the moment he releases the coconut until it hits the pavement is 190% longer than the time it takes for the impact to reach you. The angle (measured with respect to the ground ) at which you are viewing the monkey when he drops the nut is 5 degrees. Assumimg the speed of sound is constant 343m/s, from what height did the monkey drop the nut?
Thanks

 

[berkeman]

How would you start working on this problem? Draw a sketch of the tree, monkey, coconut, 5 degrees, etc. What kinematic equation will tell you how long it takes the coconut to hit the ground? What sound velocity and distance equation should you use to tell you how long it takes the coconut impact sound to reach you? Then what?

 

[kyle8921]

for your question! This is a great problem to practice using kinematics equations. To solve for the coconut drop height, we can use the following kinematics equation:

y = y0 + v0t + 1/2at^2

where y is the final height of the coconut, y0 is the initial height (in this case, the height of the monkey's position in the tree), v0 is the initial velocity (in this case, the coconut is dropped from rest so v0 = 0), t is the time elapsed, and a is the acceleration due to gravity (9.8 m/s^2).

First, we need to find the time it takes for the impact to reach you. The problem states that this time is 190% longer than the time it takes for the impact to reach you, so we can set up the following equation:

t = (1.9)t0

where t0 is the time it takes for the impact to reach you. We can rearrange this equation to solve for t0:

t0 = t/1.9

Next, we need to find the distance between the monkey and you. This can be calculated using trigonometry, since we are given the angle at which you are viewing the monkey. Using the tangent function, we can find the distance (d) as follows:

d = tan(5 degrees) * y0

Now, we can plug in our values for t0, d, and a into the kinematics equation to solve for y0, the height at which the monkey dropped the coconut:

y0 = y - 1/2at0^2 - v0t0
y0 = 0 - 1/2(9.8 m/s^2)(t/1.9)^2 - 0
y0 = -0.258t^2

Finally, we can use the speed of sound to calculate the time it takes for the impact to reach you (t0):

d = v0t0
t0 = d/v0
t0 = (tan(5 degrees) * y0)/343 m/s
t0 = (-0.258t^2)/343 m/s
t0 = -0.000753t^2 seconds

Now, we can plug this value for t0 back into our equation for y0 to solve for the height at which the monkey dropped the coconut:

y0