You and your friend are leaning over the edge of the roof of a tall building. At the same instant you throw a baseball straight up with an initial speed vo and your friend throws a marble straight down with the same initial velocity vo. Assume that you can ignore air resistance. (1) Which object will reach the ground first, the baseball or the marble? (2) How will their speeds compare just before hitting the ground?

To determine which object will reach the ground first, we need to compare their respective trajectories and analyze their motion. Let's break it down step by step:

1. Determine the motion of the baseball:
- When you throw the baseball straight up with an initial velocity vo, it will experience upward acceleration due to gravity, which will gradually slow it down.
- As the baseball reaches its maximum height, its vertical velocity will reduce to zero, and it will start to fall downwards.
- The time it takes for the baseball to reach its maximum height will be the same as the time it takes to fall back to the ground with the same magnitude of initial velocity.

2. Determine the motion of the marble:
- Your friend throws the marble straight down with the same initial velocity vo.
- The marble, being thrown downwards, will experience downward acceleration due to gravity, which will increase its speed over time.

Now, let's answer the questions step by step:

(1) Which object will reach the ground first, the baseball or the marble?
- Both the baseball and the marble are subject to gravity, and their motion is independent of the initial velocity they were thrown with.
- Since they have the same initial velocity, both the baseball and the marble will take the same time to reach the ground.
- Therefore, both objects will reach the ground at the same time.

(2) How will their speeds compare just before hitting the ground?
- Just before hitting the ground, both the baseball and the marble will have the same magnitude of speed.
- This is because their acceleration due to gravity will be the same, and they would have fallen for the same amount of time.
- Therefore, their speeds will be equal when they hit the ground.

In summary, both the baseball and the marble will reach the ground at the same time, and their speeds will be equal just before hitting the ground.

To answer these questions, let's break down the motion of both objects. We'll use the standard kinematic equations to analyze their paths.

1) Which object will reach the ground first?

The baseball is thrown straight up, while the marble is thrown straight down. The key to determining which object reaches the ground first lies in their initial velocities and the gravitational acceleration.

The baseball thrown upwards will experience gravitational acceleration pulling it downward, slowing it down until it reaches its highest point, where its velocity becomes zero. Then, gravity will act in the opposite direction, accelerating it downward.

The marble, on the other hand, is already moving downward, so gravity will accelerate it further in the same direction.

Since both objects start with the same initial velocity vo and experience the same gravitational acceleration, their paths will be symmetrical. Therefore, the time it takes for each object to reach the ground will be the same. They will hit the ground simultaneously.

2) How will their speeds compare just before hitting the ground?

Just before hitting the ground, both objects will have been accelerated by gravity for the same amount of time. Since their initial velocities are the same, and they have undergone the same amount of acceleration due to gravity, their final velocities just before hitting the ground will also be the same.

In conclusion, both the baseball and the marble will reach the ground at the same time, and their speeds just before hitting the ground will also be equal.

They both pass you at speed vo on the way down, but the baseball passes you later due to wasting time going up first.

Therefore the marble hits first.
Their motion is identical all the way down after passing you on the way down, same speed at every altitude.