Example: The faster fall

Understanding the situation

Two dense objects (so air drag can be ignored) are shot straight up at the same time from the same height. Object A is shot with a speed of 1 m/s, object B with a speed of 2 m/s. 

Presenting a sample problem

A. Which takes longer to come back to its starting point?

1. 1. Object A
   2. Object B
   3. Both take the same.
   4. I can’t tell since you didn’t give me the masses.
   5. I can’t tell for some other reason.

B. Two dense objects (so air drag can be ignored) are shot up at the different angles at same time from the same height. They follow the trajectories shown. Which will hit its target first?

1. 1. Object A
   2. Object B
   3. Both take the same.
   4. I can’t tell since you didn’t give me the masses.
   5. I can’t tell for some other reason.

Solving this problem

A. Object B

At first glance this seems difficult. Object B will go higher and so will travel a longer distance, but it's going faster. These facts act in opposite directions, the first suggesting B would take more time, the second suggesting B would take less time. 

You can do the calculation and see that the result is object B. But it's easy to figure out from thinking about what happens. Each projectile slows as it goes up until its velocity is cut to 0, then it falls. The velocity falling just reverses what happened going up, so either projectile will take the same amount of time going down as going up. So let's just look at the falling part. Both start at 0 velocity and start to speed up. Both will go through the same velocity pattern until they reach 1 m/s. At that time, projectile A will have hit the ground but projectile B will still have a ways to fall, both since it started higher and since it still has to accelerate from 1 m/s up to 2 m/s before it hits the ground.

B. Object B

At first glance this also looks confusing. A goes higher, but B travels farther. Will the two results cancel? What we learned from our analysis of free-fall in flat-earth gravity is that the sideways and up-and-down motions are independent. What determines when it hits the ground is only the up-and-down motion: when y=0 it's on the ground again. From the previous problem we learned that the higher it goes the longer it takes to come back to the ground, therefore, since B does not go as high as A, it will hit first.

Joe Redish 2/4/19