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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Two balls are thrown upward from the edge of the cliff in Example 7. The first is thrown with a speed of 48 ft/s and the other is thrown a second later with a speed of 24 ft/s. Do the balls ever pass each other?

To determine

To find:

The required time for the two balls to pass each other.

Explanation

1) Concept:

i. If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is Fx+c, where c is an arbitrary constant.

ii.

αt=ddxvt

iii.

vt=ddxst

2) Calculations:

Here, two balls are thrown upward from the edge of the cliff above the ground.

The motion is vertical and we choose the positive direction to be upward.

At time t  the distance above the ground is s(t) and the velocity v(t) is decreasing. Therefore, the acceleration must be negative.

αt=dvdt=-32

αt=-32

Taking the general antiderivative of αt by using the rules of antiderivative is

vt=-32t+v0, using v0(initial velocity) as a constant term

Now, take general antiderivative of vt, using the rules of antiderivative is

st=-32×t22+v0t+s0, using s0 (initial position) as a constant term

 st=-16t2+v0t+s0

Now, the first ball is thrown with a speed of 48 feet/s

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