Two Peaches, Two Seconds, One Roof: A peach is thrown straight up from the edge of the roof of a building. A second peach is dropped from the roof 2s later.

a. If the height of the building is 60m what must the initial speed of the first peach be if both are to hit the ground at the same time?
b. Now consider the same situation, but now let the initial speed vi of the first peach be given and treat the height as an unknown h. What must the height of the building be for both peach to reach the ground at the same time for each of the following velocities:
i. vi = 13 m/s ii. vi = 19.2 m/s
c. If vi is greater than some value vmax than a value of h does not exist that allows both peaches to hit the ground at the same time. Solve for vmax. The vmax has some physical interpretation, what is it?
d. If vi is lesser than some value vmin than a value of h does not exist that allows both peach to hit the ground at the same time. Solve for vmin. The vmin has some physical interpretation, what is it?

a. To determine the initial speed of the first peach, we need to use the kinematic equations of motion. We have two objects in free fall: the first peach that is thrown up and the second peach that is dropped. For both peaches to hit the ground at the same time, their time of flight must be the same.

Let's consider the first peach that is thrown up. The time of flight can be calculated using the equation:

t = (2 * initial velocity) / acceleration

Since the peach is thrown up, the acceleration is negative (due to gravity), and the height of the building is given as 60m. We can assume that the final height of the first peach is zero, and the initial height is 60m.

Using the equations of motion:

0 = 60 + (initial velocity * t) + (0.5 * acceleration * t^2)

Since the peach is moving against gravity, the acceleration is -9.8 m/s^2.

Substituting the value of t from the first equation into the second equation, we can solve for the initial velocity of the first peach. This will be the speed required for both peaches to hit the ground simultaneously.

b. In this case, the initial speed of the first peach is given, and we need to find the height of the building for both peaches to hit the ground at the same time. We can start by using the equation of motion:

h = (initial velocity * t) + (0.5 * acceleration * t^2)

For both peaches to hit the ground at the same time, we set t = 2 seconds for the second peach, as mentioned in the question.

We can substitute the values of initial velocity, acceleration, and t into the equation to solve for h. We repeat this process for each given velocity to find the corresponding height of the building.

c. To find the maximum initial velocity (vmax) for which there is no value of h that allows both peaches to hit the ground at the same time, we can analyze the scenario.

For the second peach, which is dropped from the roof, the time of flight is simply 2 seconds. However, the time of flight for the first peach depends on its initial velocity.

Using the equation t = (2 * initial velocity) / acceleration, we find that if the initial velocity of the first peach is greater than some value vmax, the time of flight becomes less than 2 seconds. In this case, the peaches cannot hit the ground simultaneously.

Solving for vmax would provide the limiting initial velocity beyond which a value of h does not exist.

The physical interpretation of vmax is that it represents the maximum initial velocity for the first peach for which both peaches can hit the ground at the same time.

d. Similarly, to find the minimum initial velocity (vmin) for which there is no value of h that allows both peaches to hit the ground at the same time, we can analyze the scenario.

Using the same equation t = (2 * initial velocity) / acceleration, we find that if the initial velocity of the first peach is smaller than some value vmin, the time of flight becomes greater than 2 seconds. In this case, the peaches cannot hit the ground simultaneously.

Solving for vmin would provide the limiting initial velocity below which a value of h does not exist.

The physical interpretation of vmin is that it represents the minimum initial velocity for the first peach for which both peaches can hit the ground at the same time.