Question

During the first half of a 3.0 mile bike race with a friend of yours, your friend bikes the whole way at a constant speed of 23 mi/hr. You start from rest and accelerate constantly from the beginning. You both are neck-and-neck at the half-way point of the race. For the second half of the race, you maintain the same speed you had at the half-way point. How much faster does your friend need to bike in order to tie with you at the end of the race (assume that they pick up their pace very quickly and so you can think of them as moving at a constant speed for pretty much the entire second half of the race)?

Answer 1

In the first half way of 3.0 mile bike race i.e. 1.5 mile, your friend bikes the whole way with 23mi/hr speed.
Therefore your friend takes time

from the begining of race.

Now you starts to accelerate from the begining with zero initial speed and according to the question you get your friend after crossing half way that means after 0.0652 hr.

Suppose d is the distance covered by you and your friend at a acceleration after t time with zero initial speed.
To calculate the acceleration we can use kinematic equations as follows


let with this acceleration you achieve a speed of v after 0.0652 hr and v will be


Therefore to maintain you side by side for the last half way your friend has to increase his/her bike speed from 23 mi/hr to 46 mi/hr very quickly as you set your speed constant at 46 mi/hr.

So, now your friend needs to faster his/her bike (46 - 23) = 23 mi/hr more than his/her initial speed.