A stone is thrown upward and rises 9ft before starting to fell. Find the total time taken for the stone to return to the starting point

To find the total time taken for the stone to return to the starting point, we can break down the problem into two parts: the upward journey and the downward journey.

The upward journey:
When the stone is thrown upward, it rises 9 ft. We can use the formula for the distance covered during free fall to calculate the time it takes for the stone to reach the highest point of its trajectory. The formula is:

d = ut + (1/2)gt^2

where:
d = distance covered
u = initial velocity (which is the velocity at the start, before the stone begins to fall)
t = time taken
g = acceleration due to gravity (approximately 32 ft/s^2)

In this case, the stone rises 9 ft, so the distance covered is 9 ft. The initial velocity is the velocity at the highest point, before the stone starts to fall. Since the velocity at the highest point is 0 ft/s (because the stone momentarily stops before reversing its direction), we can set u = 0.

And since the stone is only moving in the vertical direction, we can ignore horizontal motion and only consider the gravitational acceleration.

Plugging these values into the formula, we have:

9 = 0t + (1/2)(32)t^2

Rearranging the equation, we get:

4t^2 = 9

t^2 = 9/4

t = √(9/4) = 3/2 = 1.5 seconds

Therefore, it takes 1.5 seconds for the stone to reach the highest point of its trajectory.

The downward journey:
Now, to find the total time, we need to consider the time it takes for the stone to fall from the highest point back to the starting point. The time taken for the downward journey is the same as the time taken for the upward journey.

So, the total time taken for the stone to return to the starting point is:

Total time = Upward journey time + Downward journey time
= 1.5 seconds + 1.5 seconds
= 3 seconds

Therefore, the total time taken for the stone to return to the starting point is 3 seconds.

To find the total time taken for the stone to return to the starting point, we need to consider the time taken for the stone to reach its maximum height and then the time taken for it to fall back down.

First, let's determine the time taken to reach the maximum height. We can use the equation for the height of an object in freefall:

h = v₀t + (-1/2)gt²

Where:
h is the height
v₀ is the initial velocity
t is the time
g is the acceleration due to gravity

Since the stone is thrown upward and rises 9ft, its initial velocity, v₀, is positive. The acceleration due to gravity, g, is negative since it acts in the opposite direction of the stone's motion.

In this case, the initial height, h, is 9ft, and we can assume the initial velocity, v₀, is 0ft/s (because when the stone reaches its maximum height, its velocity becomes 0).

So the equation for the stone's height becomes:

9 = 0t + (-1/2)gt²

Simplifying the equation, we get:

9 = (-1/2)gt²

To find the time it takes for the stone to reach its maximum height, we can plug in the value of g, the acceleration due to gravity, which is approximately -32ft/s²:

9 = (-1/2)(-32)t²

Rewriting the equation, we get:

9 = 16t²

Divide both sides by 16:

9/16 = t²

Take the square root of both sides:

√(9/16) = t

Simplifying, we get:

t ≈ ±0.75s

Since time cannot be negative, we take the positive value:

t ≈ 0.75s

So it takes approximately 0.75 seconds for the stone to reach its maximum height.

Next, to find the time taken for the stone to fall back down, we can use the same equation as before:

h = v₀t + (-1/2)gt²

This time, the initial height, h, is 0ft (since the stone is falling back to the starting point), and the initial velocity, v₀, is 0ft/s (since the stone starts from rest when it falls back down).

The equation becomes:

0 = 0t + (-1/2)gt²

Simplifying, we get:

0 = (-1/2)gt²

Dividing both sides by -1/2, we get:

0 = gt²

Since the acceleration due to gravity, g, is approximately -32ft/s², the equation becomes:

0 = -32t²

Simplifying, we get:

0 = t²

This equation indicates that when time is 0, the stone is at its initial position, which is the starting point.

Therefore, it takes 0 seconds for the stone to fall back down and return to the starting point.

To find the total time taken, we add the time taken to reach the maximum height and the time taken to fall back down:

Total time = time to reach maximum height + time to fall back down
Total time ≈ 0.75s + 0s
Total time ≈ 0.75s

So, the total time taken for the stone to return to the starting point is approximately 0.75 seconds.