That at some time during the race, the acceleration of both the cars are same.

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 4, Problem 19P
To determine

That at some time during the race, the acceleration of both the cars are same.

Expert Solution

Explanation of Solution

Given information:

In an automobile race along a straight road, car A passes car B twice.

Calculations:

Let SA(t) and SB(t) be the position functions of cars A and B resp.

Let F(t)=SA(t)SB(t)

Now it is given that A passes B twice, therefore there exist (at least) three times when their positions are equal.

Let, t1 is the time when A overtakes B the first time,

t2 is the time when B overtakes A, and

t3 is the time when A again overtakes B.

Now position of the cars at time t1 , t2 and t3 is,

F(t1)=F(t2)=F(t3)

By the mean value theorem, there exists a time u1 such that, t1<u1<t2 ,where F'(u1)=0 and similarly, a time t2<u2<t3 such that F'(u2)=0 .

Applying the mean value theorem again, this time to the function F'(t) , we see that there is a time r , u1<r<u2 ,such that F''(r)=0 .

Now, F''(t)=SA''(t)SB''(t)=0 ,

so SA''(t)=SB''(t)

Thus we can see that the acceleration of both the cars are same at some time r .

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