A stone is shot upward with a speed of 24.4 m/s from a tower and lands at the base of the tower with a speed of 44.0 m/s. How long does it take?

do the fall first

v = g t
44 = 9.81 t
t = 4.49 seconds falling

now the up part
Vi = 24.4
v = Vi - g t
0 = 24.4 - 9.81 t where t is time rising
t = 2.49 seconds upward
so
4.49 + 2.49 seconds

To find the time it takes for the stone to land at the base of the tower, we can analyze the problem using the equations of motion. Let's assume the upward direction as positive and the downward direction as negative.

We are given:
Initial upward velocity, u = +24.4 m/s
Final downward velocity, v = -44.0 m/s

We need to find the time, t.

Using the equation of motion:
v = u + at

where:
v is the final velocity,
u is the initial velocity,
a is the acceleration, and
t is the time.

Since the stone is shot upward and then falls, the acceleration due to gravity, g, will act downward. The magnitude of the acceleration is the acceleration due to gravity, g = 9.8 m/s^2.

For the upward motion:
v = 0 m/s (at the highest point)

0 = 24.4 - 9.8t
9.8t = 24.4
t = 24.4 / 9.8
t = 2.49 seconds (approx.)

For the downward motion:
v = -44.0 m/s

-44.0 = 0 - 9.8t (since the stone is falling downwards, the initial velocity is 0 m/s)

9.8t = 44.0
t = 44.0 / 9.8
t = 4.49 seconds (approx.)

The total time taken by the stone to go up and come down is the sum of the upward and downward times, t_total = t_upward + t_downward:
t_total = 2.49 + 4.49
t_total = 6.98 seconds (approx.)

Therefore, the stone takes approximately 6.98 seconds to reach the base of the tower.

To find the time it takes for the stone to go up and come back down, we can use the fact that the initial vertical velocity of the stone is 24.4 m/s and the final vertical velocity is -44.0 m/s (negative value because it's going downward). We need to find the time it takes to reach this final velocity.

The equation that relates initial velocity, final velocity, and time for an object in free fall is:

v = u + at,

where:
v = final velocity,
u = initial velocity,
a = acceleration (in this case, acceleration due to gravity),
t = time.

In this case, we can assume the acceleration due to gravity is -9.8 m/s^2 (negative because it's in the opposite direction to the initial velocity).

So, let's first find the time it takes for the stone to go up:

We have:
v = -44.0 m/s,
u = 24.4 m/s,
a = -9.8 m/s^2,
t = ? (to be found).

Using the formula, we have:
-44.0 = 24.4 + (-9.8)t.

Rearranging the equation to solve for t, we get:
t = (44.0 - 24.4) / (-9.8).

Calculating this expression gives t = 1.84 seconds (rounded to two decimal places).

Since it takes the same amount of time to come back down, the total time for the stone to go up and down is twice that value.

Therefore, the total time it takes for the stone to go up and come back down to the base of the tower is 2 * 1.84 seconds = 3.68 seconds.