If a ball is thrown straight up into the air with an initial velocity of 60 ft/s...?

Question

If a ball is thrown straight up into the air with an initial velocity of 60 ft/s, its height in feet after t seconds is given by y=60t−16t^2. Find the average velocity for the time period beginning when t=1 and lasting:

 

(i) 0.5 seconds:

 

(ii) 0.1 seconds:

 

(iii) 0.01 seconds:

 

(iv) 0.0001 seconds:

 

Finally, based on the above results, guess what the instantaneous velocity of the ball is when t=1.
Answer:

Answer 1

s(t) = 60t -16t^2

v(t) = 60-32t

v(1) = 28 ft/sec instantaneous speed at time t=1

v(1.5)= 60-48= 12 ft/sec

(28+12)/2 = 40/2 = 20 ft/sec average speed

average velocity =d/t= [s(1,5)-s(1)]/.5 = 60(1.5)-16(1.5)^2 - 60+16 over 2 = 90-36-60+15 = 9/.5 = 18 ft/sec.

in 1 second it's moved 44 feet, in 1.5 seconds it moved 54 feet. It moved 10 feet in half a second. 10/1/2 = 20 feet per second average speed

the shorter the time traveled the closer to 28 ft/sec is the average speed

v(1.1) = 60-32(1.1) = 60-35.2 = 24.8

(28+24.8)/2 = 52.8/2 = 26.4 ft/sec average speed

v(1.01) = 60-32(1.01) = 60-33.32 = 26.68

28+26.68 = 54.68 = 27.34 ft/sec average speed

v(1.0001) = 60-32(1.0001) = 60 - 32.0032 = 27.9968

28+27.9968)/2 = 55.9968/2 = 27.9984 ft/sec average speed

instantaneous speed at t=1 is 28 ft/sec

Answer 2

s(t) = 60t -16t^2

v(t) = 60-32t

v(1) = 28 ft/sec instantaneous speed at time t=1

v(1.5)= 60-48= 12 ft/sec

(28+12)/2 = 40/2 = 20 ft/sec average speed

average velocity =d/t= [s(1,5)-s(1)]/.5 = 60(1.5)-16(1.5)^2 - 60+16 over 2 = 90-36-60+15 = 9/.5 = 18 ft/sec.

in 1 second it's moved 44 feet, in 1.5 seconds it moved 54 feet. It moved 10 feet in half a second. 10/1/2 = 20 feet per second average speed

the shorter the time traveled the closer to 28 ft/sec is the average speed

v(1.1) = 60-32(1.1) = 60-35.2 = 24.8

(28+24.8)/2 = 52.8/2 = 26.4 ft/sec average speed

v(1.01) = 60-32(1.01) = 60-33.32 = 26.68

28+26.68 = 54.68 = 27.34 ft/sec average speed

v(1.0001) = 60-32(1.0001) = 60 - 32.0032 = 27.9968

28+27.9968)/2 = 55.9968/2 = 27.9984 ft/sec average speed