Two stones dropped at the same time. Quick question, .

[wolfmanzak]

Homework Statement

Two identical stones are dropped from a tall building, one after the other. Assume air resistance is negligible. While both stones are falling, what will happen to the vertical distance between them?

Homework Equations

a) It will increase.

b) It will first increase and then remain constant.

c) It will remain the same.

d) It will decrease.

The Attempt at a Solution

Assuming air resistance in negligible, wouldn't the stones both be affected equally and fall to the ground at the same rate of gravitational pull? They would both fall at the rate of 9.81m/s^2 and thus the distance between them would remain the same? Or would it increase until both were falling at this rate and then remain equal?

The book says that the answer is "It will increase" but that would mean that two things being affected by the same force have different accelerations? Why is this?

 

[Doc Al]

The key phrase is 'one after the other'. They are not dropped at the same time.

 

[Hootenanny]

Assuming air resistance in negligible, wouldn't the stones both be affected equally and fall to the ground at the same rate of gravitational pull?

The would experience the same acceleration, yes.

They would both fall at the rate of 9.81m/s^2

Correct.

and thus the distance between them would remain the same?

No.

Or would it increase until both were falling at this rate and then remain equal?

No.

Suppose the first stone was released at $t=0$ from a height of $h=h_0$. Its height is then governed by

$$h_1 = h_0 -\frac{g}{2}t^2$$

Yes? Suppose the second stone is released from the same height at $t=t_0>0$. The its height is

$$h_2 = h_0 - \frac{g}{2}(t-t_0)^2\;\;\;\text{for}\;\;\;t>t_0$$.

The difference in height is then

$$h_2-h_1 = -\frac{g}{2}(t-t_0)^2 + \frac{g}{2}t^2$$

which obviously increases with time (you can plot it to convince yourself).

More intuitively, since the first stone starts accelerating before the second stone and both accelerate at the same rate, the first stone is always traveling faster than the second.

Does that make sense?

Edit: I see Doc beat me to it

 

[wolfmanzak]

I think I understand the concept now. It makes a little more sense after these explanations. Thank you both.

 

[rwisz]

I would like to clarify that the distance between the two stones will indeed increase as they fall. This is due to the fact that, although both stones are affected by the same gravitational force, they also have their own individual masses. As a result, they will have different accelerations, with the heavier stone having a slightly greater acceleration. This means that the distance between the two stones will increase as they fall, but it will do so at a slower rate than if they were falling at the same acceleration. This is a result of Newton's second law, which states that the force applied to an object is proportional to its mass and its acceleration. Therefore, the heavier stone will experience a greater force and thus a greater acceleration. However, as the stones continue to fall, their speeds will increase and eventually they will both reach the same acceleration. At this point, the distance between them will remain constant until they reach the ground.