A stone is thrown straight up from the edge of a roof, 600 feet above the ground, at a speed of 12 feet per second.
A. Remembering that the acceleration due to gravity is -32 feet per second squared, how high is the stone 6 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?
h = -16t^2 + 12t + 600
a) sub in t=6
b) set -16t^2 + 12t + 600 = 0
divide by -4
4t^2 - 3t - 150 = 0
use the formula to find t = .... (use the positive answer only)
c) v = -32t + 12
plug in your answer from b)
h(t)=hi+vi(t)-1/2 * 32*t^2 put t=6 and solve.
now use the same equation, solve for t when h(t)=0. Ignore the t negative answer.
vf=vi(t)+gt put in tfinal, and solve, OR
final KE=initialKE+initialGPE
or vf^2=vi^2+2(32)(600)
A. Well, ignoring the fact that stones aren't the most aerodynamic objects, let's calculate how high the stone is 6 seconds later. Using the formula for distance, we know that the height (h) can be calculated using the formula h = initial height + initial velocity × time + 0.5 × acceleration × time^2. Since we are calculating the height after 6 seconds, let's plug in the values: initial height = 600 feet, initial velocity = 12 feet per second, acceleration = -32 feet per second squared, and time = 6 seconds. Crunching the numbers, we get the final height of the stone. But honestly, the stone won't be too happy about it because it's just going to come right back down.
B. Ah, the ground is the ultimate destination for all falling objects. To find the time the stone hits the ground, we can use the equation h = initial height + initial velocity × time + 0.5 × acceleration × time^2 (again). This time, we set the height (h) to zero, as that's the height of the ground. Plugging in the values we have: initial height = 600 feet, initial velocity = 12 feet per second, acceleration = -32 feet per second squared, and solving for time, we'll find out when the stone decides to give gravity a big ol' hug.
C. When the stone finally embraces the ground and decides to settle down, its velocity will be a little bit alarming! But hey, let's calculate it anyway. We can use the equation for velocity, which is velocity = initial velocity + acceleration × time. Setting initial velocity to 12 feet per second, acceleration to -32 feet per second squared, and using the time we found in part B, we'll determine the velocity of the stone right before it says hello to the ground. It'll be quite a memorable landing, I assure you.
A. To determine the height of the stone 6 seconds later, we need to consider the motion of the stone under the influence of gravity.
First, let's determine the equation for the height of the stone as a function of time. The equation for position in vertical motion under constant acceleration is given by:
h(t) = h₀ + v₀t + 1/2at²
Where:
h(t) is the height at time t
h₀ is the initial height (600 feet)
v₀ is the initial velocity (12 feet per second)
a is the acceleration due to gravity (-32 feet per second squared)
t is the time in seconds
Plugging in the values:
h(t) = 600 + 12t - 16t²
To find the height after 6 seconds, substitute t = 6 into the equation:
h(6) = 600 + 12(6) - 16(6)²
Simplifying the equation:
h(6) = 600 + 72 - 16(36)
h(6) = 600 + 72 - 576
h(6) = 96 feet
Therefore, the stone is 96 feet high 6 seconds later.
B. To determine the time at which the stone hits the ground, we need to find when the height equals zero (h(t) = 0).
Using the equation for height:
600 + 12t - 16t² = 0
To solve this quadratic equation, we can factor it or use the quadratic formula. Factoring the equation:
4t(4 - t) = 0
Setting each factor equal to zero:
4t = 0 or 4 - t = 0
Solving for t:
t = 0 (This is the initial time when the stone is thrown)
4 - t = 0
t = 4
Therefore, the stone hits the ground at t = 4 seconds.
C. To find the velocity of the stone when it hits the ground, we can use the equation for velocity:
v(t) = v₀ + at
Plugging in the values:
v(t) = 12 + (-32)(4)
Simplifying the equation:
v(t) = 12 - 128
v(t) = -116 feet per second
Thus, the velocity of the stone when it hits the ground is -116 feet per second. The negative sign indicates that the velocity is directed downward.