A 0.26 kg rock is thrown vertically upward from the top of a cliff that is 32 m high. When it hits the ground at the base of the cliff the rock has a speed of 28 m/s.

(a) Assuming that air resistance can be ignored, find the initial speed of the rock.

(b) Find the greatest height of the rock as measured from the base of the cliff.

To find the initial speed of the rock, we can use the equations of motion. We'll assume that the positive direction is upward.

(a) Finding the initial speed:
We can use the equation for the final velocity of an object in free fall:

v_final = v_initial + a * t,

where:
v_final = final velocity (28 m/s),
v_initial = initial velocity (unknown, what we want to find),
a = acceleration due to gravity (-9.8 m/s^2),
t = time taken for the rock to hit the ground.

Given the acceleration due to gravity and knowing that the object is in free fall, we can determine the time it takes for the rock to fall from the top of the cliff to the base:

Using the equation of motion for displacement:

s = ut + (1/2) * a * t^2,

where:
s = displacement (32 m, the height of the cliff),
u = initial velocity (unknown),
t = time.

Rearranging the equation, we have:

t^2 - (2s/a) * t + (2u/a) = 0.

Plugging in the values, we can solve this quadratic equation for t. Since the time cannot be negative, we will take the positive root.

Using the quadratic formula, we have:

t = (-b + √(b^2 - 4ac)) / 2a,

where:
a = 1,
b = -(2s/a) = -2s/g,
c = (2u/a) = (2u/g).

Plugging in the values, we get:

t = (-(-(2s/g)) + √((-(2s/g))^2 - 4 * 1 * (2u/g))) / 2 * 1.

Simplifying this equation:

t = (2s/g + √((4s^2/g^2) - 8u/g)) / 2.

Now, we can calculate the time it takes for the rock to fall to the ground.

Once we have the value of t, we can substitute it back into the first equation to find the initial velocity (v_initial) of the rock.

(b) Finding the greatest height:
To find the highest point reached by the rock, we can use the equation for the final velocity at any given point during the upward journey:

v_final = v_initial + a * t.

At the highest point, the final velocity is zero (v_final = 0). By solving this equation for t, we can determine the time it takes for the rock to reach its highest point.

Using the value of t, we can substitute it into the equation for displacement during upward motion:

s = ut + (1/2) * a * t^2.

Plugging in the values, we can solve for the displacement (height) at the highest point.

Using these steps, we can find the initial speed of the rock and the greatest height it reaches.