A stone is dropped from the edge of a roof, and hits the ground with a velocity of −170 feet per second. Assume the acceleration due to gravity is -32 feet per second squared. How high (in feet) is the roof?
I have found s which is 16^2 but I do not know what formula to use.
distance = -16t^2 + h
velocity = -32t
but we know that is -170
so -32t = -170
t = 85/16 seconds
so how far his it fallen in 85/16 seconds?
distance = 16(85/16)^2
= 7225/16 ft or
appr 451.56 ft
Well, to solve this problem, we can use the kinematic equation for vertical motion:
s = ut + (1/2)at^2
Where s is the height, u is the initial velocity, a is the acceleration due to gravity, and t is the time.
In this case, the stone is dropped from rest, so the initial velocity (u) is 0 ft/s. The acceleration due to gravity (a) is -32 ft/s^2. The time (t) is the time it takes for the stone to hit the ground, so we need to find that.
Let's use the formula for final velocity (v) to solve for time. We know that the final velocity (v) is -170 ft/s, the initial velocity (u) is 0 ft/s, and the acceleration (a) is -32 ft/s^2. The formula is:
v = u + at
Substituting the values, we have:
-170 = 0 + (-32)t
Now let's solve for t:
-170 = -32t
Dividing both sides by -32:
t = -170 / -32
t ≈ 5.3125 seconds
Now that we have the time, we can substitute it back into the first equation to find the height (s):
s = ut + (1/2)at^2
s = 0 + (1/2)(-32)(5.3125)^2
s = 0 + (-16)(5.3125)^2
s = -16(28.25)
s ≈ -452 feet
Now, since height can't be negative, let's take the absolute value:
|s| ≈ 452 feet
So, the height of the roof is approximately 452 feet. Keep in mind that negative velocities and heights indicate the direction of motion, but we disregard the negative symbol when finding the magnitude (or absolute value) of the quantity.
Hope that puts a smile on your face while you calculate the height of that roof!
To find the height of the roof, you can apply the equation of motion for an object in free fall. The equation is:
s = ut + (1/2)gt^2
Where:
s = height
u = initial velocity
g = acceleration due to gravity
t = time
In this case, the stone is dropped, so the initial velocity is zero (u = 0). The acceleration due to gravity is -32 feet per second squared (g = -32). We need to find the time it takes for the stone to hit the ground, and the height will be negative since it is measured below the starting point.
The equation becomes:
s = (1/2)gt^2
The final velocity (v) is given as -170 feet per second. Using the formula for final velocity in free fall:
v = u + gt
Since the final velocity is -170 feet per second, the initial velocity (u) is 0, and the acceleration due to gravity is -32 feet per second squared, we can solve for time (t):
-170 = 0 + (-32)t
t = -170 / (-32)
t = 5.3125 seconds
Now we can substitute this value of time into the equation for height:
s = (1/2)(-32)(5.3125^2)
s = (1/2)(-32)(28.2406)
s = -16(-28.2406)
s = 451.8509 feet
Therefore, the height of the roof is approximately 451.8509 feet.
To determine the height of the roof, you can use the equation of motion for an object in free fall:
s = u*t + (1/2)*a*t^2
Where:
- s is the distance or height traveled by the object (in this case, the height of the roof).
- u is the initial velocity of the object (in this case, 0 ft/s as the stone is dropped).
- t is the time taken by the stone to reach the ground (unknown).
- a is the acceleration due to gravity (-32 ft/s^2).
Since the stone is dropped, the initial velocity (u) is 0 ft/s. The final velocity of the stone when it hits the ground is -170 ft/s. The negative sign indicates the downward direction.
We can rearrange the equation to solve for the time (t):
s = (1/2)*a*t^2
t^2 = (2s) / a
t = sqrt((2s) / a)
Now plug in the values:
a = -32 ft/s^2
u = 0 ft/s
v = -170 ft/s
Using the formula for time, we can calculate t:
t = sqrt((2s) / a)
t = sqrt((2 * s) / (-32))
Now, the final velocity (v) is given by:
v = u + a*t
-170 = 0 - 32*t
Solve for t:
-170 = -32t
t = (-170) / (-32)
t = 5.3125 seconds (approximately)
Now we can use the time and the equation of motion to find the height (s) of the roof:
s = u*t + (1/2)*a*t^2
s = 0 * 5.3125 + (1/2)(-32)(5.3125)^2
s = (1/2)(-32)(5.3125)^2
Finally, calculate s to find the height of the roof.