A stone is thrown straight down from the edge of a roof, 750 feet above the ground, at a speed of 7 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?
To find the height of the stone at 5 seconds later, we need to use the formula for displacement under constant acceleration:
Displacement = Initial Velocity * Time + 0.5 * Acceleration * Time^2
A. For the first question, we are given that the stone is thrown downward, so the initial velocity is -7 feet per second (negative because it is in the opposite direction to the positive direction of measurement). The acceleration due to gravity is -32 feet per second squared. Plugging in the values into the formula:
Displacement = -7 * 5 + 0.5 * (-32) * (5^2)
Displacement = -35 + 0.5 * (-32) * 25
Displacement = -35 - 16 * 25
Displacement = -35 - 400
Displacement = -435
Since the stone is thrown downward, the displacement is negative. Therefore, the height of the stone 5 seconds later is 435 feet below the initial height of 750 feet. We subtract 435 feet from 750 feet to get the final height:
Final Height = 750 - 435
Final Height = 315 feet
B. To find the time it takes for the stone to hit the ground, we need to set the displacement equal to zero and solve for time:
Displacement = -7 * t + 0.5 * (-32) * t^2 = 0
Simplifying the equation:
-7t - 16t^2 = 0
We can factor out a t:
t(-7 - 16t) = 0
This equation will be zero when either t = 0 or -7 - 16t = 0. Since time cannot be negative, we can solve the second equation for t:
-7 = 16t
t = -7/16
Since time cannot be negative, we discard this solution. Therefore, the only solution is t = 0. This means that the stone hits the ground at time t = 0, or immediateaneously.
C. Finally, to calculate the velocity of the stone when it hits the ground, we can use the formula:
Final Velocity = Initial Velocity + (Acceleration * Time)
Plugging in the values:
Final Velocity = -7 + (-32 * 0)
Final Velocity = -7
The negative sign indicates that the velocity is in the opposite direction of the initial velocity. Therefore, the velocity of the stone when it hits the ground is 7 feet per second, but in the opposite direction in which it was thrown.