While standing at the edge of the roof of a building, you throw a stone upward with an initial speed of 6.93 m/s. The stone subsequently falls to the ground, which is 17.1 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 = -9.8 m/s2.

While standing at the edge of the roof of a building, you throw a stone upward with an initial speed of

7.17 m/s. The stone subsequently falls to the ground, which is 14.7 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 = 9.8 m/s2. (This is NOT a suggestion to carry out such an experiment!)

Well, I hate to break it to you, but throwing stones off buildings is not a wise choice. It tends to make people grumpy and could land you in a bit of a legal pickle. But since you threw it already, let's figure this out!

To find the speed at which the stone impacts the ground, let's break it down. When the stone is at the peak of its motion, its vertical velocity will be 0 m/s because it's about to change direction. Using the equation v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time, we can find the time it takes for the stone to reach the highest point.

Since the stone is thrown upwards, the acceleration due to gravity will be -9.8 m/s². Plugging in the values, we have:

0 = 6.93 m/s + (-9.8 m/s²)t

Now, let me calculate that.

Calculating...

Calculating...

Ah, so t = 0.707 seconds. This will be the time it takes for the stone to reach the highest point.

Now, to find the total time the stone is in the air, we'll double this time because it takes the same amount of time for the stone to go up and come back down. So, the total time in the air is approximately 2 * 0.707 = 1.414 seconds.

To find the speed at which the stone impacts the ground, we need to use the equation v = u + at again, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time. This time, the initial velocity is 0 m/s because the stone starts falling downwards.

Let's calculate that:

v = 0 m/s + (-9.8 m/s²)(1.414 s)

Calculating...

Calculating...

Ah, the stone will impact the ground with a speed of approximately 13.8 m/s (rounded to one decimal place). Ouch! That's not going to tickle!

Remember, though, this is assuming there's no air resistance and gravity is a steady -9.8 m/s². Keep it safe and leave the stone-throwing antics to the clowns like me!

To solve this problem, we can use the equations of motion for projectile motion.

Step 1: Find the time the stone takes to reach its maximum height.

Let's assume the initial height is h = 0. The height of the building is given as 17.1 m. The final height is also 0 (since the stone falls back to the ground).

The equation for the height of an object in projectile motion is:

h = h0 + v0t + (1/2)gt^2

For the stone at the maximum height, h = 17.1 m, h0 = 0, v0 = 6.93 m/s, and g = -9.8 m/s².

Plugging in these values, we get:

17.1 = 0 + 6.93t - (1/2)(9.8)t^2

Rearranging the equation, we have a quadratic equation:

-4.9t^2 + 6.93t - 17.1 = 0

Solving this quadratic equation, we find two possible values for t. Since we are interested in the time it takes for the stone to reach its maximum height, we will consider only the positive value.

Step 2: Find the time it takes for the stone to fall back to the ground.

For the stone to fall back to the ground, the final height is 0, the initial height is 17.1 m, and g = -9.8 m/s².

Using the same equation as before:

0 = 17.1 + 6.93t + (1/2)(-9.8)t^2

Rearranging the equation, we have:

4.9t^2 + 6.93t + 17.1 = 0

Solving this quadratic equation, we find two possible values for t. Since we are interested in the time it takes for the stone to fall back to the ground, we will consider only the positive value.

Step 3: Find the time the stone is in the air.

The total time the stone is in the air is the sum of the time it takes to reach the maximum height and the time it takes to fall back to the ground.

t_total = t_max_height + t_fall

Step 4: Find the speed at which the stone impacts the ground.

The speed with which the stone impacts the ground can be calculated using the equation:

v = v0 + gt

where v0 is the initial velocity, g is the acceleration due to gravity, and t is the time of impact.

Let's calculate the values step-by-step.

Step 1: Find the time the stone takes to reach its maximum height.

Using the quadratic formula, the positive value of t is approximately 0.707 seconds.

Step 2: Find the time it takes for the stone to fall back to the ground.

Using the quadratic formula, the positive value of t is approximately 1.468 seconds.

Step 3: Find the time the stone is in the air.

t_total = 0.707 + 1.468 = 2.175 seconds (approximately)

Step 4: Find the speed at which the stone impacts the ground.

Using the equation v = v0 + gt, with v0 = 6.93 m/s, g = -9.8 m/s², and t = t_total = 2.175 seconds, we can calculate the speed:

v = 6.93 - 9.8 * 2.175 = -12.97 m/s

Since speed cannot be negative, we take the magnitude of the velocity and get that the speed at which the stone impacts the ground is approximately 12.97 m/s.

To find the speed at which the stone impacts the ground, we can first find the time it takes for the stone to reach the ground. Since we are ignoring air resistance, the only force acting on the stone is gravity, which causes it to accelerate downward at a rate of -9.8 m/s^2.

We can use the kinematic equation for displacement to find the time it takes for the stone to fall to the ground:

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

In this equation, s is the displacement (17.1 m), u is the initial velocity (in this case, 0 m/s since the stone is momentarily at rest at the topmost point), a is the acceleration (-9.8 m/s^2), and t is the time we want to find.

Substituting the values into the equation:

17.1 = 0*t + (1/2)(-9.8)(t^2)
17.1 = -4.9t^2
t^2 = 17.1/-4.9
t^2 = 3.49
t ≈ √3.49
t ≈ 1.87 seconds

So the time it takes for the stone to fall to the ground is approximately 1.87 seconds.

To find the speed at which it impacts the ground, we can use the equation for velocity:

v = u + at

In this equation, v is the final velocity we want to find, u is the initial velocity (6.93 m/s upwards), a is the acceleration (-9.8 m/s^2), and t is the time (1.87 seconds) we found earlier.

Substituting the values into the equation:

v = 6.93 + (-9.8)(1.87)
v ≈ 6.93 - 18.33
v ≈ -11.4 m/s

Since the velocity is negative, it means the stone impacts the ground in the downward direction. Therefore, the speed at which the stone impacts the ground is approximately 11.4 m/s.

h = ho + -(Vo^2)/2g

h = 17.1 + 6.93^2/19.6 = 19.6 m. Above
gnd.

a. V^2 = Vo^2 + 2g*h = 0 + 19.6*19.6 =
382.2
V = 19.6 m/s.

b. Tr = -Vo/g = -6.93/-9.8 = 0.707 s. =
Rise time.

h = 0.5g*t^2 = 19.6 m.
4.9t^2 = 19.6
t^2 = 4
Tf = 2 s. = Fall time.

Tr+Tf = 0.707 + 2 = 2.707 s = Time in air.