The position function of an object moving along a straight line is given by s =

Question

The position function of an object moving along a straight line is given by s = f(t). The average velocity of the object over the time interval [a, b] is the average rate of change of f over [a, b]; its (instantaneous) velocity at t = a is the rate of change of f at a. A ball is thrown straight up with an initial velocity of 96 ft/sec, so that its height (in feet) after t sec is given by s = f(t) = 96t 16t2. (a) What is the average velocity of the ball over the following time intervals? [3,4] Incorrect: Your answer is incorrect. ft/sec [3,3.5] Incorrect: Your answer is incorrect. ft/sec [3,3.1] Incorrect: Your answer is incorrect. ft/sec (b) What is the instantaneous velocity at time t = 3? Incorrect: Your answer is incorrect. ft/sec

Explanation / Answer

Average Velocity = Total Displacement/Time Interval

(a) [3,4]

Average Velocity = f(4) - f(3)/( 4 - 3 )

f(4) = 96(4) - 16(42) = 128

f(3) = 96(3) - 16(32) = 144

=> Average Velocity = (128 - 144)/(4 - 3) = -16 ft/s

(b) [3,3.1]

Average Velocity = f(3.1) - f(3)/( 3.1 - 3 )

f(3.1) = 96(3.1) - 16(3.12) = 143.84

f(3) = 96(3) - 16(32) = 144

=> Average Velocity = (143.84 - 144)/(3.1 - 3) = -1.6 ft/s

(c) [3,3.5]

Average Velocity = f(3.5) - f(3)/( 3.5 - 3 )

f(3.1) = 96(3.5) - 16(3.52) = 140

f(3) = 96(3) - 16(32) = 144

=> Average Velocity = (140 - 144)/(3.5 - 3) = -9 ft/s

(d) Instantaneous Velocity = s'(t) = 96 - 32t

s'(3) = 96 - 32(3) = 0 ft/s

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