**Solution:**

To solve this problem, we can use the equations of motion for objects in free fall. The equations of motion are:

**1.** *s = ut + 0.5 a t^2*
**2.** *v = u + at*
**3.** *v^2 = u^2 + 2as*

Where:
- *s* is the distance traveled by the object
- *u* is the initial velocity of the object
- *v* is the final velocity of the object
- *t* is the time taken
- *a* is the acceleration of the object

In this case, both objects are dropped from rest, so their initial velocities (*u*) are both zero.

Let's calculate the time taken for both objects to fall to the ground using equation (1):

For the first object:
*s1 = 150 m*

*s1 = ut + 0.5 a t^2*
*150 = 0 + 0.5 * 10 * t^2*
*150 = 5t^2*
*t^2 = 30*
*t = √30*

For the second object:
*s2 = 100 m*

*s2 = ut + 0.5 a t^2*
*100 = 0 + 0.5 * 10 * t^2*
*100 = 5t^2*
*t^2 = 20*
*t = √20*

So, both objects take different times to reach the ground. Now, let's calculate the difference in their heights after 2 seconds using equation (1) again:

For the first object:
*s1 = ut + 0.5 a t^2*
*s1 = 0 + 0.5 * 10 * (2)^2*
*s1 = 0 + 0.5 * 10 * 4*
*s1 = 0 + 20*
*s1 = 20 m*

For the second object:
*s2 = ut + 0.5 a t^2*
*s2 = 0 + 0.5 * 10 * (2)^2*
*s2 = 0 + 0.5 * 10 * 4*
*s2 = 0 + 20*
*s2 = 20 m*

The difference in their heights after 2 seconds is the same for both objects, which is 20 meters.

Now, let's discuss how the difference in heights varies with time:

From our calculations, we can see that the difference in heights remains constant at 20 meters after 2 seconds. This is because both objects experience the same acceleration due to gravity (10 m/s^2) and they are dropped from rest. As a result, their velocities and distances covered will be the same at any given time. Therefore, the difference in their heights will always remain constant.

In summary, the difference in heights between two objects dropped from rest at different heights and with the same acceleration will remain constant throughout their fall.

The difference in there heights would be same i.e 50m as both will travel same distance in a certain time whatever it is.The difference in their height remains same with time.