A stone is thrown straight up from the edge of a roof, 725 feet above the ground, at a speed of 10 feet per second.
A. Remembering that the acceleration due to gravity is -32 feet per second squared, how high is the stone 5 seconds later?
B. At what time does the stone hit the ground?
C. What is the velocity of the stone when it hits the ground?
by this time you should know that the height and velocity functions will be
h = 725 + 10t - 16t^2
v = 10 - 32t
and you should be able to handle this little gem.
To solve these problems, we will use the kinematic equations of motion. The first equation we'll use is:
v = u + at
Where:
v = final velocity
u = initial velocity
a = acceleration
t = time
Let's solve the problems step by step:
A. How high is the stone 5 seconds later?
To find the height of the stone 5 seconds later, we need to find the final displacement. We can use the second equation of motion:
s = ut + (1/2)at^2
Where:
s = displacement or height
u = initial velocity
t = time
a = acceleration
Given:
u = 10 feet per second
t = 5 seconds
a = -32 feet per second squared (negative because the stone is moving upwards)
Plugging in the values, we get:
s = (10)(5) + (1/2)(-32)(5)^2
s = 50 - 400
s = -350 feet
Since the displacement is negative, it means the stone is 350 feet below the starting point. Therefore, the height of the stone 5 seconds later is 725 - 350 = 375 feet.
B. At what time does the stone hit the ground?
To find the time when the stone hits the ground, we need to find the time when the height is zero. We can use the same equation as in part A, but set s = 0:
0 = (10)t + (1/2)(-32)t^2
This is a quadratic equation in the form of at^2 + bt + c = 0, where:
a = -16
b = 10
c = 0
Solving this equation, we get two solutions: t = 0 and t = 5.
Since the stone was thrown upwards, the time t = 0 is the initial time when the stone was thrown. Therefore, we take the positive value t = 5 as the time when the stone hits the ground.
C. What is the velocity of the stone when it hits the ground?
To find the velocity of the stone when it hits the ground, we can use the first equation of motion:
v = u + at
Given:
u = 10 feet per second (initial velocity)
t = 5 seconds (time when the stone hits the ground)
a = -32 feet per second squared (acceleration)
Plugging in the values, we get:
v = 10 + (-32)(5)
v = 10 - 160
v = -150 feet per second
The negative sign indicates that the stone is traveling downwards. Therefore, the velocity of the stone when it hits the ground is 150 feet per second downward.