A ball is thrown vertically upward from the ground at an initial velocity of 64 ft/sec. After t seconds, the height of the ball is given by the displacement (height) function s(t) = 64t − 16t^2 feet.

Question

(a) Find the velocity function of the ball at time t. v(t) = s(t).

(b) What is the velocity at time t = 3 seconds.

(c) How long will it take the ball to reach its highest point?

(d) How high will the ball reach?

(e) How long will it take the ball to hit the ground after it is thrown?

Answer 1

gravitational pull=32.2f/s^2;

b. original speed=64ft/sec up. After 2 sec, 64-32.2=31.8=95.8ft; 3sec,31.8-32.2=-.4+95.8=95.4ft.

c. 2.9seconds

d. 95.4ft, see#b

e. Falling distance=(t^2+t)/2*32.2.

t=1.99 seconds

+2.9=4.89

Answer 2

gravitational pull=32.2f/s^2;

b. original speed=64ft/sec up. After 2 sec, 64-32.2=31.8=95.8ft; 3sec,31.8-32.2=-.4+95.8=95.4ft.

c. 2.9seconds

d. 95.4ft, see#b

e. Falling distance=(t^2+t)/2*32.2.

t=1.99 seconds

+2.9=4.89

Answer 3

The equation for the displacement s(t) as a function of time t is s(t) = 64t - 16t². This is a height above the ground where we assume the upward motion originated.

The initial velocity of the ball is 64 ft/s

(a) The velocity is the first time derivative of the displacement: v(t) = ds/dt = 64 - 32t. We are just taking the first time derivative of the function s(t).

(b) v(t = 3) = 64 - 32(3) = -32 ft/s. The negative sign indicates that the ball is moving downward at this time.

(c) When the ball reaches its highest point, it momentarily pauses before falling. We can find the time for the ball to reach the highest point by setting the velocity equal to zero, to represent this pause.

v(t) = 64 - 32t = 0

or 32t = 64

or t = 2 sec.

(d) The maximum height of the ball occurs at t = 2 sec, as calculated in part (c).

s(t = 2) = 64(2) - 16(2)²

or s(t = 2) = 64 feet

(e) We can use symmetry. It we know that it took 2 seconds for the ball to reach its maximum height, then it will take another 2 seconds for the ball to fall back to the ground. So, the total time of flight is 4 sec.