Which of the following changes would result in the greatest deltatime?

[needingtoknow]

Homework Statement

Several balls of different masses and radii are being dropped from a balcony. Why would increasing the height from which the balls are dropped so taht each ball reaches its terminal velocity before hitting the ground increase the time difference between the time the first ball hits the ground and the time the last ball hits the ground?

The Attempt at a Solution

The solutions say to achieve the largest time difference, each ball needs to achieve the greatest difference in acceleration, which means each should achieve its maximum drag force. Since drag force depends on velocity and velocity increases until it reaches terminal velocity, then the drag force is maximized when the velocity reaches the terminal velocity. Each object needs a long enough fall to achieve terminal velocity.

I don't understand why each ball needs to achieve the greatest difference in acceleration, in order to achieve the greatest time difference between the first and last ball that strikes the ground. Can someone clear this confusion? Thank you.

 

[BiGyElLoWhAt]

If you drop a ball and a hammer off of the leaning tower of pisa, like that one guy did that one time, what would you expect to happen? What's the acceleration of each object? If $\Sigma F = ma$ then it should follow that $a = \frac{\Sigma F}{m}$

Look up the equation for drag, and assume that the balls (presumably being made of the same material) have the same drag coefficient. Now, if you drop both of those balls at the same time, it makes sense that the farther the second one is behind the first one will hit at a later time, right? Think about the relationship between acceleration and displacement w.r.t. time.

 

[needingtoknow]

I understand that as time passes, displacement of each will increase under a constant acceleration, but how does show why each ball needs to achieve the greatest difference in acceleration, in order to achieve the greatest time difference between the first and last ball that strikes the ground?

 

[BiGyElLoWhAt]

take it to the extremes. Let both balls drop at the same time. First situation, they both have an acceleration of 100 m/s^2. What's the time difference between the balls hitting the ground? Now similar situation, but 1 of the balls is accelerating at 50 ms^-2. What's the time difference now, (say they fall a distance of 100m)?
Now for the extreme, let one of the balls still accelerate 100m/s^2 and the other 0 m/s^2.

If you really wanted to, you could go through and math all this out with calculus, but this question seems pretty conceptual, and if you can understand it enough to explain it, I think that should be enough.

 

[HalfLostAndSearching]

Increasing the height from which the balls are dropped would result in the greatest deltatime because it allows each ball to reach its terminal velocity before hitting the ground. This means that each ball will experience the maximum difference in acceleration as it falls, leading to a longer time difference between the first and last ball hitting the ground. This is because the drag force, which is dependent on velocity, will increase until it reaches terminal velocity, and a longer fall allows for a longer period of time for this to occur. Therefore, increasing the height of the drop would result in the greatest deltatime.