A stone is thrown straight up from the edge of a roof, 700 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 2 seconds later? Don't forget to enter correct units.
(B). At what time does the stone hit the ground?
(C).What is the velocity of the stone when it hits the ground?
To answer these questions, we can use the equation of motion for an object in free fall under constant acceleration due to gravity. The equation is:
s = ut + (1/2)at^2
where s is the displacement, u is the initial velocity, a is the acceleration, and t is the time.
(A) To find the height of the stone 2 seconds later, we substitute the given values into the equation:
s = 0 + (1/2)(-32)(2)^2
s = 0 + (1/2)(-32)(4)
s = 0 + (-16)(4)
s = -64 feet
Since the stone was thrown upwards, the negative sign indicates that the height is below the starting point of 700 feet. Therefore, the stone is at a height of 700 - 64 = 636 feet.
(B) To find the time when the stone hits the ground, we need to find when the displacement (s) becomes zero. Since the stone was thrown upwards, the acceleration due to gravity is acting in the opposite direction, causing the stone to fall. Setting s = 0, we solve for t:
0 = (1/2)(-32)t^2
0 = -16t^2
This equation has two solutions, t = 0 and t = sqrt(0), which means that the stone hits the ground at t = 0 or t = 0 seconds. Since we are considering the stone after it was thrown, we can disregard t = 0. Therefore, the stone hits the ground at t = sqrt(0) seconds, which simplifies to t = 0 seconds.
(C) To find the velocity of the stone when it hits the ground, we can use another equation of motion:
v = u + at
where v is the final velocity.
Since the stone was thrown upwards, the initial velocity u is positive 10 feet per second. Substituting the values into the equation:
v = 10 + (-32)(0)
v = 10 feet per second
Therefore, the velocity of the stone when it hits the ground is 10 feet per second, but in the opposite direction of the initial velocity since the stone falls downwards.