A ball is thrown upward with a speed of 48 ft/s from the edge of a bridge 32 ft above the ground. The acceleration due to gravity is about 32 feet per second per second.

Question

Find the ball’s height above the ground 𝑡 seconds later.

When does the ball reach its maximum height?

When does the ball hit the ground?

On what interval of time is the speed of the ball increasing? What about the velocity? Explain.

Answer 1

first calculate HMAX as ball is thrown upward.

mgh + 0.5 mu2 =mgH +0.5(o)²

gh + 0.5 u2 =gH

HMAX = 32 + [(0.5*48*48)/32]

=68 ft

Now calculate the velocity of the ball before falling

v²=2gH

v²=(2*32*68)

v²=4352 {square root both sides}

v=66.000 ft/sec

Answer 2

The equation for height is h(t) = -16t^2 + vt + ho

where v is the initial velocity at time t=0 and ho is the initial height

h(t) = -16t^2 + 48t + 32

to find max height, take the derivative, set = 0, and solve for t

h'(t) = -32t + 48 = 0

t=48/32 = 3/2 or 1.5 seconds

the ball hits the ground when h=0

h(t) = -16t^2 + 48t + 32 = 0

divide by -16

h = t^2 -3t -2 = 0

complete the square or use the quadratic formula

t^2 -3t + 9/4 = 2 + 9/4

(t-3/2)^2 = 17/4

t-3/2 = + or - (1/2) square root of 17

ignore the negative square root

t = 1.5 + 2.06 = 3.56 seconds approximately to hit the ground

the ball is increasing in speed only from it's maximum height to the time it hits the ground

for an interval of about 3.56-1.5 = 2.06 seconds

The velocity generally is the speed. the interval of increasing speed or increasing velocity is the same,

from t=1.5 to t = 3.56, a little over 2 seconds, from maximum height where speed = 0 to the time

it hits the ground,

Unless you want to define speed as the absolute value of velocity. Then velocity can be negative or positive, and never increases, since its positive velocity decreases and then become negative, decreasing further.

Then speed is always positive and decreases from t=0 to t=1.5. At t=0 speed is 48 ft/sec, then 1 1/2 seconds later speed drops to zero, then speed increases but in the opposite direction until it hits the ground.