How Long Until the Second Car Overtakes the First?

[skoomafiend]

Homework Statement

A car starts from rest and accelerates uniformly at 3 m/s2. A second car starts from rest 6 s later at the same point and accelerates uniformly at 5 m/s2. How long does it take the second car to overtake the first car?
How would I solve this? If someone can break it down step by step, it would be great.

Homework Equations

x=x₀ + v₀t + 1/2at²

The Attempt at a Solution

I tried solving it by using the above formula and solving for t. Didnt quite work out.

car 1's time = car two's time +6 and then solving for t.

 

[pr0blumz]

Find the position of the first car. Then, use that distance and set it equal to the position equation of the second car to find the time.

 

[skoomafiend]

I don't understand what you are saying, can you clarify please? How would I find the distance with the information given?

 

[pr0blumz]

Find the velocity of the first

 

[skoomafiend]

find the position of the first car in relation to what? the initial 6 second lead?

 

[pr0blumz]

Correct, the position with the 6 second lead

 

[skoomafiend]

(3/2)t²= (5/2)(t- 6)²

I've tried solving for this, doesn't seem to match that answer given at the back of the book.
Am I doing something wrong?

 

[pr0blumz]

What's the answer? Where are you getting those numbers?

 

[skoomafiend]

a₁=3m/s²
a₂=5m/s²

t₂=(t₁- 6) ...t₁ is the time for the first car, then t₂ left 6 seconds after the car, so it had ( t₁- 6 ) seconds to get to the same position (at which point it would overtake car₁)

x = x₀ + v₀t + 1/2at₁²
x = 0 + 0 + 1/2(3)t₁²
x = (3/2)t₁²

and for car two it is..

x = 0 + 0 + 1/2(5)( t₁ - 6 )²
x = (5/2)( t₁ - 6 )²

so that means that x would be the same distance, given these times

(3/2)t₁² = (5/2)( t₁ - 6 )²

The answer at the back of the book is 22.7 s
That is not the answer I get when I solve for this.