What Is the Minimum Time for a Sportscar to Travel 1/2 Mile from Standstill?

[ttja]

Homework Statement

a A sportscar can accelerate uniformly to 120 mi/h in 30s. Its maximum braking rate cannot exceed 0.7g. what is the minimum time required to go 1/2 mi, assuming it begins and ends at rest?

Homework Equations

I drew a graph of v(t) vs t. where the initial acc. goes up to a certain time, t1, then decelerate to a rest at t2.

The Attempt at a Solution

acc= 4 mi/min
V1 = 4t

For convenience i drew another graph for when the car dec. Therefore:

acc= -0.7g
V2 = 4t1 - 0.7gt

Integrate:

d = 0.5 = int ( 4t, t, 0, t1) + int (4t1 - 0.7gt, t, 0, t2)

=> 2t1^2 + 4t1t2 - 0.7g/2 t2^2

Here is where i got lost. i know that t = t1 + t2 but i don't know how to derive t with its components to get the minimum time. please help and thank you.

 

[hotvette]

Interesting problem. You can determine t₂ in terms of t₁ because the velocity must be zero in the end. Other considerations are less than max deceleration and coast time between acceleration and deceleration.