While standing at the edge of the roof of a building, you throw a stone upward with an initial speed of 6.11 m/s. The stone subsequently falls to the ground, which is 12.7 m below the point where the stone leaves your hand. At what speed does the stone impact the ground? How much time is the stone in the air? Ignore air resistance and take g = 9.81 m/s2. (This is NOT a suggestion to carry out such an experiment!)

hf=hi+vi*t-4.86t^2

hi=0 hf=-12.7
solve for t.
speed:
vf=vi-9.8t
solve for vf

To determine the speed at which the stone impacts the ground and the time it remains in the air, we can use the equations of motion for projectile motion.

Let's break down the problem step by step:

Step 1: Analyze the motion of the stone.

The stone is thrown upward, slows down, reaches the highest point, and then falls downward due to gravitational acceleration. We can consider this as two separate motions: upward motion and downward motion.

Step 2: Calculate the time taken to reach the highest point.

We can use the kinematic equation for vertical motion to determine the time it takes for the stone to reach the highest point:

Δy = v₀y * t + (1/2) * a * t²

Where:
Δy is the vertical displacement (12.7 m)
v₀y is the initial velocity in the vertical direction (6.11 m/s)
a is the acceleration due to gravity (-9.81 m/s²)
t is the time taken to reach the highest point

Plugging in the values, we get:

12.7 m = 6.11 m/s * t + (1/2) * (-9.81 m/s²) * t²

Simplifying the equation, we have:

-4.9 t² + 6.11 t - 12.7 = 0

Now, we can solve this quadratic equation for t using the quadratic formula:

t = (-b ± √(b² - 4ac)) / (2a)

a = -4.9
b = 6.11
c = -12.7

After solving the quadratic equation, we will get two values for t. However, we will only consider the positive value since time cannot be negative in this context.

Step 3: Calculate the time of flight.

The total time the stone remains in the air is twice the time it takes to reach the highest point. So, we multiply the value of t by 2 to determine the total time of flight.

Step 4: Calculate the speed at impact.

To calculate the speed at which the stone impacts the ground, we can use the equation for vertical motion:

v = v₀y + a * t

Where:
v is the final velocity of the stone at impact (to be determined)
v₀y is the initial vertical velocity (6.11 m/s)
a is the acceleration due to gravity (-9.81 m/s²)
t is the time taken to reach the ground

Substituting the values, we get:

v = 6.11 m/s + (-9.81 m/s²) * t

Using the value of t obtained earlier, we can calculate the final velocity at impact.

By following these steps, we can find the speed at which the stone impacts the ground and the time it remains in the air.