if you throw a ball up, it reaches its apex at 100ft at 1.5 secs, how do you find initial velocity and height?

Question

Using the formula h(t) = -1/2gt^2 + vt+s, and where "v" is initial velocity, "s" is initial height, "t" is time, and g is 32 ft/sec/sec.

Also, after 4 seconds it hits the ground, and you are standing on a building.

Answer 1

Here's your original position function of the ball:
h(t) = -1/2gt^2 + vt+s

Here's the first derivative of that function:
h'(t) = V(t) = -32t + v

After taking the derivative, you now have the velocity function
Here, i'm just showing that, at 1.5 seconds, the velocity is zero.
V(1.5) = 0 ft/s

V(1.5) = 0 = -32(1.5) + v

v = 32*1.5 --------------> v = 48 ft/s

Since it hits the ground after 4 seconds,

h(4) = 0 = -1/2(32)(4^2) + 4(48) + s

s = 1/2(32)(4^2) - 4(48)

s = 64 ft

Answer 2

Work the problem in reverse.

From h(t) = -1/2gt^2 + vt+s

h(t) = - 1/2(32 ft/sec/sec (1.5 sec)^2) + 0(1.5) + 100 feet
= -1/2 (32 X 2.25) + 0 + 100 feet
= - 1/2 (72) + 100 = - 36 + 100 = 64 feet

The velocity would be = g(t) = 32ft/sec/sec (1.5 sec) = 48 ft/sec.

The first 3.0 seconds (1.5 sec up + 1.5 sec down) puts the ball back at the original height. It will be 64 feet up and traveling at -48 ft/sec (negative because it is going down).

The height at 4 seconds = 0 feet = -1/2 (32 ft/sec/sec (1.0 sec)^2) - 48 ft/sec(1.0 sec) + 64 feet

at t = 0, the ball is at 64 feet with a velocity of 48 ft/sec
at t = 1.5, the ball is at 100 feet with a zero velocity
at t = 3.0, the ball is at 64 feet with a velocity of -48 ft/sec
at t = 4.0, the ball is at 0 feet

You should be able to calculate the impact velocity if you need it.

Answer 3

First figure out the initial velocity of the ball. Since the ball reaches it apex after 1.5 seconds, its upward velocity is zero at this point. Therefore, it took gravity 1.5 seconds to reduce the balls speed to zero. Since the velocity of an object after t seconds in flight is given by vi-g*t, the initial upward velocity of the ball was 48 ft/sec. Knowing this allows you to solve for initial height; simply plug in 100 for h(t), 1.5 for t and 48 for v initial and you can solve for s.

Answer 4

initial velocity is 48m/s.

initial position is 64 ft