A stone is dropped from the top of a cliff and one second latter a second stone is thrown vertically downward with a velocity of 20m/s. How far below the top of the cliff will the stone overtake the first?

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To find out how far below the top of the cliff the second stone will overtake the first stone, we first need to determine the time it takes for the first stone to reach the same vertical position as the second stone.

We can start by calculating the time it takes for the first stone to fall to the point where the second stone was thrown from. The equation for the distance fallen by an object in free fall is given by:

s = ut + (1/2)at^2

where:
s = distance fallen
u = initial velocity (which is 0 for a falling object)
t = time
a = acceleration due to gravity = 9.8 m/s²

In this case, the distance fallen (s) is equal to the height of the cliff. We can use the equation to calculate the time it takes for the first stone to fall to that height.

Let's assume the height of the cliff is h meters.

Using the equation, we have:
h = (1/2)gt^2

Rearranging the equation to solve for time t, we get:
t = √(2h/g)

Now, we know that the second stone is thrown 1 second later with an initial velocity (u) of 20 m/s.

The equation for the distance covered by an object in free fall is given by:
s = ut + (1/2)at^2

For the second stone, the distance covered (s) is the height of the cliff. We can use the equation to calculate the time taken for the second stone to fall the same distance as the first stone.

s = 20t + (1/2)gt^2

Substituting the value of t we obtained earlier, we can solve for s.

s = 20√(2h/g) + (1/2)g(√(2h/g))^2

This simplifies to:
s = 20√(2h/g) + (1/2)(2h)

Now we can find the difference in height between the two stones. The second stone overtakes the first when this difference is zero.

Let's calculate the difference in height:
Difference = (20√(2h/g) + (1/2)(2h)) - h

Simplifying the equation, we get:
Difference = 20√(2h/g) + h

To find how far below the top of the cliff the second stone overtakes the first, we need to calculate the value of h that makes the Difference equal to zero.

This equation can be solved using algebraic methods or by using numerical methods such as estimation or a graphing calculator.