Question

A stone is dropped from the upper observation deck of a tower, 250 m above the ground. (Assume g = 9.8 m/s2.) (a) Find the distance (in meters) of the stone above ground level at time t. h(t) = (b) How long does it take the stone to reach the ground? (Round your answer to two decimal places.) s (c) With what velocity does it strike the ground? (Round your answer to one decimal place.) m/s (d) If the stone is thrown downward with a speed of 2 m/s, how long does it take to reach the ground? (Round your answer to two decimal places.) s

Answer 1

(a) $y(t)=250 - 4.9 t^2$

For an object in free-fall, the vertical position at time t is given by:

$y(t) = h + ut - \frac{1}{2}gt^2$

where

h is the initial vertical position

u is the initial vertical velocity

g = 9.8 m/s^2 is the acceleration of gravity

t is the time

In this problem,

h = 250 m

u = 0 (the stone starts from rest)

So, the vertical position of the stone is given by

$y(t) = 250 - \frac{1}{2}(9.8) t^2 = 250 - 4.9 t^2$

(b) 7.14 s

The time it takes for the stone to reach the ground is the time t at which the vertical position of the stone becomes zero:

y(t) = 0

Which means

$y(t) = h - \frac{1}{2}gt^2=0$

So for the stone in the problem, we have

$250 - 4.9 t^2 = 0$

Solving for t, we find:

$t=\sqrt{\frac{250}{4.9}}=7.14 s$

(c) -70.0 m/s (downward)

The velocity of an object in free fall is given by the equation

$v(t) = u - gt$

where

u is the initial velocity

g = 9.8 m/s^2 is the acceleration of gravity

t is the time

Here we have

u = 0

So if we substitute t = 7.14 s, we find the velocity of the stone at the time it reaches the ground:

$v=0-(9.8 m/s^2)(7.14 s)=-70.0 m/s$

The negative sign means the direction of the velocity is downward.

(d) 6.94 s

In this situation, the stone is thrown downward with an initial speed of 2 m/s, so its initial velocity is

u = -2 m/s

So the equation of the vertical position of the stone in this case is

$y(t) = h + ut - \frac{1}{2}gt^2=250 - 2t - 4.9 t^2$

By solving the equation, we find the time t at which the stone reaches the ground.

We find two solutions:

t = -7.35 s

t = 6.94 s

The first solution is negative, so it has no physical meaning, therefore we discard it. So, the time it takes for the stone to reach the ground is:

t = 6.94 s