a stone is thrown vertically upwards fron the edge of a cliff of height 100m, with a velocity of 9m/s. The stone then fall to the ground(the foot of the cluff). Determine the total time the stone will take to hit the ground

V = Vo + g*t.

Tr = (V-Vo)/g = (0-9)/-9.8 = 0.918 s.

Tf1 = Tr = 0.918 s. = Fall time from hmax to edge of cliff.

h = Vo*t + 0.5g*t^2 = 100m.
9*t + 4.9t^2 = 100
4.9t^2 + 9t - 100 = 0.
Use Quad. Formula.
Tf2 = 3.69 s. = Fall time from edge of cliff to Gnd.

T = Tr + Tf1 + Tf2.
T = 0.918 + 0.918 + 3.69 = 5.53 = Total
time in flight.

To determine the total time the stone will take to hit the ground, we need to consider two parts of the stone's motion: the upward motion and the downward motion.

First, let's determine the time it takes for the stone to reach its maximum height during the upward motion.

We can use the equation for displacement in uniformly accelerated motion:

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

where s is the displacement, u is the initial velocity, t is the time, and a is the acceleration.

In this case, the stone is thrown vertically upwards, so the initial velocity is +9 m/s (positive direction) and the acceleration is -9.8 m/s^2 (due to gravity, acting opposite to the direction of motion).

The displacement in the upward motion is the height of the cliff, which is 100 m.

So, we can write the equation as:

100 = (9)t + (1/2)(-9.8)t^2.

Simplifying this equation gives us a quadratic equation:

-4.9t^2 + 9t - 100 = 0.

We can solve this equation to find the time it takes for the stone to reach its maximum height. Using the quadratic formula, we get:

t = (-b ± √(b^2 - 4ac)) / (2a),

where a = -4.9, b = 9, and c = -100.

Solving this equation gives us two possible values for t: t = 5 s and t = -4 s. Since time cannot be negative, we discard the negative solution.

Therefore, the time it takes for the stone to reach its maximum height during the upward motion is 5 seconds.

Now, let's determine the time it takes for the stone to fall from its maximum height to the ground during the downward motion.

Since the stone is falling freely under the influence of gravity, we can use the equation for freely falling objects:

s = ut + (1/2)gt^2,

where s is the displacement, u is the initial velocity, t is the time, and g is the acceleration due to gravity.

In this case, the initial velocity is 0 m/s (since the stone starts falling from rest at the top of its trajectory), the acceleration due to gravity is +9.8 m/s^2 (acting downwards), and the displacement is the same as the initial height of the cliff, which is 100 m.

So, we can write the equation as:

100 = (1/2)(9.8)t^2.

Simplifying this equation gives us:

4.9t^2 = 100.

Solving for t gives us:

t^2 = 100 / 4.9,

t = √(100 / 4.9),

t ≈ 4.04 s.

Therefore, the time it takes for the stone to fall from its maximum height to the ground is approximately 4.04 seconds.

Finally, to determine the total time, we add the time it takes to reach the maximum height during the upward motion (5 seconds) and the time it takes to fall from the maximum height to the ground during the downward motion (4.04 seconds):

Total time = 5 s + 4.04 s = 9.04 seconds.

Therefore, the stone will take approximately 9.04 seconds to hit the ground.