When a ball is thrown straight down from the top of a tall building, with initial velocity 30 ft/sec, the distance from the release point at time t in seconds is given by s(t)=16t^2+30t.
If the release point is 300 ft above ground, what is the velocity of the ball at the time it hits the ground?
I think you would have to average the feet per seconds but I'm not sure.
Use the physics equation:
Vf^2=Vi^2+2ad where Vi is the initial velocity, a is the acceleration due to gravity or 9.80 m/s^2 and d equals your distance
To find the velocity of the ball at the time it hits the ground, we need to determine when the ball hits the ground first. Given the equation for the distance function:
s(t) = 16t^2 + 30t
Since the release point is 300 ft above the ground, we can set s(t) equal to 300 and solve for t:
16t^2 + 30t = 300
We can simplify the equation:
16t^2 + 30t - 300 = 0
Divide the entire equation by 2 to simplify further:
8t^2 + 15t - 150 = 0
Now we can solve this quadratic equation. Since it doesn't factor neatly, we'll use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
Where a = 8, b = 15, and c = -150. Plugging in the values:
t = (-15 ± √(15^2 - 4 * 8 * -150)) / (2 * 8)
Simplifying the equation:
t = (-15 ± √(225 + 4800)) / 16
t = (-15 ± √5025) / 16
We can use the positive solution since time cannot be negative:
t = (-15 + √5025) / 16
Calculating this value, we find:
t ≈ 2.85 seconds
So, it will take approximately 2.85 seconds for the ball to hit the ground.
To find the velocity of the ball at this time, we can differentiate the distance function with respect to time (t) to obtain the derivative, which represents the instantaneous velocity:
v(t) = s'(t) = 32t + 30
Now we can substitute the value of t when the ball hits the ground:
v(2.85) = 32(2.85) + 30
Calculating this expression:
v(2.85) ≈ 91.2 ft/sec
Therefore, the velocity of the ball at the time it hits the ground is approximately 91.2 ft/sec.
So, contrary to your assumption, we didn't need to average the feet per second. We used calculus to find the instantaneous velocity at the time of impact.