Could someone please help me with this Calculus problem? I am having a lot of trouble with it.?

Question

Had Galileo dropped a cannonball from the tower of Pisa, 179ft above the ground, the ball's height above ground t seconds into the fall would have been s=179 - 16t^2.

a. What would have been the ball's velocity at time t?

b. What would have been the ball's acceleration at time t?

c. About how long would it have taken the ball to hit the ground?

d. What would have been the ball's velocity at the moment of impact?

Answer 1

Assuming no air resistance,

s = 179 - 16t^2

a.) v = -32t First derivative
b.) a = -32 Second derivative

c.) 0 = 179 - 16t^2
16t^2 = 179
t^2 = 11.1875
t = 3.3448

d.) v(3.3448) = -32 * 3.3448 = -107.0327

You might want to check my math, it's been a semester since I last had a calc class but that should be right.

Answer 2

Don't need a lot of calculus for this one.

a) dv/dt = a = acceleration which is constant g

dv = a dt

integrate both sides V = a t + C the constant is intial velocity which is zero so V = a t

b) acceleration is constant = g = 32 ft/s^2

c) s = height

V = a t = ds/dt

ds = at dt

integrate both sides

s = (1/2) a t^2 + C

s at t = 0, is 179 ft

a = -g since direction of acceleration is opposite of height

179 = 16 t^2 solve for t when s = 0

d) V = at (from above) a = -g = 32.17 ft/sec^

plug in value for t calculated when s = 0

You're done.

Answer 3

a. velocity = first derivative = -32t
b. acceleration = 2nd derivative = -32
c. find s = 0 => t = sqrt(179/16)
d. v = -32*sqrt(179/16)

sqrt = square root

Answer 4

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