That at some time during the race, the acceleration of both the cars are same.
In an automobile race along a straight road, car A passes car B twice.
Let and be the position functions of cars A and B resp.
Now it is given that A passes B twice, therefore there exist (at least) three times when their positions are equal.
Let, is the time when A overtakes B the first time,
is the time when B overtakes A, and
is the time when A again overtakes B.
Now position of the cars at time , and is,
By the mean value theorem, there exists a time such that, ,where and similarly, a time such that .
Applying the mean value theorem again, this time to the function , we see that there is a time , ,such that .
Thus we can see that the acceleration of both the cars are same at some time .
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