(a) Answer The average speed during the first 3 seconds of fall is 14.7 m / sec . Explanation Given information: A water balloon dropped from a window high above the ground falls y = 4.9 t 2 m in t sec . Calculation: Use the formula for the speed of the object. s = Δ y Δ t Now, substitute 3 for t in the equation of path for the distance travelled after 3 seconds. y = 4.9 t 2 y = 4.9 × 3 2 y = 4.9 × 9 y = 44.1 Initially the object was at rest so y = 0 at t = 0 . Now substitute these values in formula for speed. s = Δ y Δ t = y 2 − y 1 t 2 − t 1 = 44.31 − 0 3 − 0 = 14.7 m / sec Therefore, the average speed during the first 3 seconds of fall is 14.7 m / sec . (b) To determine To find: The speed of the object at instant t = 3 sec . Answer The speed of the object at instant t = 3 sec is 29.4 m / sec . Explanation Given information: A water balloon dropped from a window high above the ground falls y = 4.9 t 2 m in t sec . Calculation: The instantaneous speed of an object at t = a seconds is the average speed from time t = a seconds to t = a + h seconds. Use the formula for average speed. s = Δ y Δ t = y 2 − y 1 t 2 − t 1 Substitute 3 + h for t in the given equation for the distance travelled. y 2 = 4.9 ( 3 + h ) 2 Substitute 3 for t in the given equation for the distance travelled. y 1 = 4.9 ( 3 ) 2 = 4.9 × 9 = 44.1 Substitute these values in the formula for average speed. s = 4.9 ( 3 + h ) 2 − 44.1 3 + h − 3 = 4.9 ( 3 + h ) 2 − 44.1 h = 4.9 ( 9 + h 2 + 6 h ) − 44.1 h Further simplify. s = 44.1 + 4.9 h 2 + 29.4 h − 44.1 h = h ( 4.9 h + 29.4 ) h = 4.9 h + 29.4 Substitute h = 0 for the exact instantaneous speed. s = 4.9 × 0 + 29.4 = 29.4 m / sec Therefore, the instantaneous speed of the object at t = 3 seconds is 29.4 m / sec .

Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)

4th Edition

ISBN: 9780133178579

Author: Ross L. Finney

Publisher: PEARSON

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Question

Chapter 2.1, Problem 69E

(a)

To determine

To find: The average speed during the first 3 seconds of fall.

(a)

Expert Solution

Answer to Problem 69E

The average speed during the first 3 seconds of fall is 14.7 m/sec .

Explanation of Solution

Given information: A water balloon dropped from a window high above the ground falls y=4.9t2 m in t sec .

Calculation:

Use the formula for the speed of the object.

s=ΔyΔt

Now, substitute 3 for t in the equation of path for the distance travelled after 3 seconds.

y=4.9t2y=4.9×32y=4.9×9y=44.1

Initially the object was at rest so y=0 at t=0 .

Now substitute these values in formula for speed.

s=ΔyΔt=y2−y1t2−t1=44.31−03−0=14.7 m/sec

Therefore, the average speed during the first 3 seconds of fall is 14.7 m/sec .

(b)

To determine

To find: The speed of the object at instant t=3 sec .

(b)

Expert Solution

Answer to Problem 69E

The speed of the object at instant t=3 sec is 29.4 m/sec .

Explanation of Solution

Given information: A water balloon dropped from a window high above the ground falls y=4.9t2 m in t sec .

Calculation:

The instantaneous speed of an object at t=a seconds is the average speed from time t=a seconds to t=a+h seconds.

Use the formula for average speed.

s=ΔyΔt=y2−y1t2−t1

Substitute 3+h for t in the given equation for the distance travelled.

y2=4.9(3+h)2

Substitute 3 for t in the given equation for the distance travelled.

y1=4.9(3)2=4.9×9=44.1

Substitute these values in the formula for average speed.

s=4.9(3+h)2−44.13+h−3=4.9(3+h)2−44.1h=4.9(9+h2+6h)−44.1h

Further simplify.

s=44.1+4.9h2+29.4h−44.1h=h(4.9h+29.4)h=4.9h+29.4

Substitute h=0 for the exact instantaneous speed.

s=4.9×0+29.4=29.4 m/sec

Therefore, the instantaneous speed of the object at t=3 seconds is 29.4 m/sec .

Chapter 2.1, Problem 68E

Chapter 2.1, Problem 70E

Chapter 2 Solutions

Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)

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