Kinematics - Motion along a straight line

[SA32]

I’m having difficulty with an assignment for my Physics class.

Here’s the question:

“Cars A and B are racing each other along the same straight road in the following manner: Car A has a head start and is a distance $$D_{A}$$ beyond the starting line at t = 0. The starting line is at x = 0. Car A travels at a constant speed $$v_{A}$$. Car B starts at the starting line but has a better engine than Car A, and thus Car B travels at a constant speed $$v_{B}$$, which is greater than $$v_{A}$$.

How long after Car B started the race will Car B catch up with Car A?
Express the time in terms of given quantities.”

My embarrassingly unsuccessful attempt:
My thinking was that, when Car B catches Car A, $$x_{A} = x_{B}$$.

I substituted the given quantities for Car A and Car B separately into the equation,
$$x = x_{o} + v_{ox}(t-t_{o}) + \frac{1} {2}*a_{x}(t-t_{o})^2$$

where:
x = position as a function of time
xo = initial position
vox = initial velocity
t = a certain time
to = initial time
ax = acceleration

For Car A, I got (simplified):
$$x = D_{A} + v_{A}(t)$$

For Car B, I got (also simplified):
$$x = v_{B}(t)$$

Equating them, $$D_{A} + v_{A}(t) = v_{B}(t)$$, and solving for t, I got:
$$\frac{v_{B}t - D_{A}} {v_{A}} = t$$

Which is incorrect.

I know I shouldn’t have “t” on the left side, but other than that I’m completely lost and would really appreciate it if anyone could point me in the right direction.

 

[SA32]

Having issues fixing the fraction in my answer, so I'll try again here.

Here is what I got:

$$\frac{v_{B}t - D_{A}} {v_{A}} = t$$

 

[Hootenanny]

It may be easier to consider their relative velocity ($v_{r} = v_{B}-v_{A}$).

 

[SA32]

Thanks a bunch, got the answer!

 

[sunbunny]

I have the same question with the cars. I've made car A equal to car B and was able to get this equation:

Da + va(t) = vb(t) and then i subtracted va(t) and then took out the common factor of t like so:

Da= t(vb-va)

then found that t= Da/(vb-va)

I'm not sure if this is the right equation, I'm pretty sure it's not so if anyone can help me that would be great!

 

[Hootenanny]

I have the same question with the cars. I've made car A equal to car B and was able to get this equation:

Da + va(t) = vb(t) and then i subtracted va(t) and then took out the common factor of t like so:

Da= t(vb-va)

then found that t= Da/(vb-va)

I'm not sure if this is the right equation, I'm pretty sure it's not so if anyone can help me that would be great!

What makes you think your answer is incorrect? If your question is the same as the one originally asked then your solution is correct.