The question is incomplete. Here is the complete question.
Cars A nad B are racing each other along the same straight road in the following manner: Car A has a head start and is a distance
beyond the starting line at t = 0. The starting line is at x = 0. Car A travels at a constant speed
. Car B starts at the starting line but has a better engine than Car A and thus Car B travels at a constant speed
, which is greater than
.
Part 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.
Part B: How far from Car B's starting line will the cars be when Car B passes Car A? Express your answer in terms of known quantities.
Answer: Part A: ![t=\frac{D_{A}}{v_{B}-v_{A}}](https://tex.z-dn.net/?f=t%3D%5Cfrac%7BD_%7BA%7D%7D%7Bv_%7BB%7D-v_%7BA%7D%7D)
Part B: ![x_{B}=\frac{v_{B}D_{A}}{v_{B}-v_{A}}](https://tex.z-dn.net/?f=x_%7BB%7D%3D%5Cfrac%7Bv_%7BB%7DD_%7BA%7D%7D%7Bv_%7BB%7D-v_%7BA%7D%7D)
Explanation: First, let's write an equation of motion for each car.
Both cars travels with constant speed. So, they are an uniform rectilinear motion and its position equation is of the form:
![x=x_{0}+vt](https://tex.z-dn.net/?f=x%3Dx_%7B0%7D%2Bvt)
where
is initial position
v is velocity
t is time
Car A started the race at a distance. So at t = 0, initial position is
.
The equation will be:
![x_{A}=D_{A}+v_{A}t](https://tex.z-dn.net/?f=x_%7BA%7D%3DD_%7BA%7D%2Bv_%7BA%7Dt)
Car B started at the starting line. So, its equation is
![x_{B}=v_{B}t](https://tex.z-dn.net/?f=x_%7BB%7D%3Dv_%7BB%7Dt)
Part A: When they meet, both car are at "the same position":
![D_{A}+v_{A}t=v_{B}t](https://tex.z-dn.net/?f=D_%7BA%7D%2Bv_%7BA%7Dt%3Dv_%7BB%7Dt)
![v_{B}t-v_{A}t=D_{A}](https://tex.z-dn.net/?f=v_%7BB%7Dt-v_%7BA%7Dt%3DD_%7BA%7D)
![t(v_{B}-v_{A})=D_{A}](https://tex.z-dn.net/?f=t%28v_%7BB%7D-v_%7BA%7D%29%3DD_%7BA%7D)
![t=\frac{D_{A}}{v_{B}-v_{A}}](https://tex.z-dn.net/?f=t%3D%5Cfrac%7BD_%7BA%7D%7D%7Bv_%7BB%7D-v_%7BA%7D%7D)
Car B meet with Car A after
units of time.
Part B: With the meeting time, we can determine the position they will be:
![x_{B}=v_{B}(\frac{D_{A}}{v_{B}-v_{A}} )](https://tex.z-dn.net/?f=x_%7BB%7D%3Dv_%7BB%7D%28%5Cfrac%7BD_%7BA%7D%7D%7Bv_%7BB%7D-v_%7BA%7D%7D%20%29)
![x_{B}=\frac{v_{B}D_{A}}{v_{B}-v_{A}}](https://tex.z-dn.net/?f=x_%7BB%7D%3D%5Cfrac%7Bv_%7BB%7DD_%7BA%7D%7D%7Bv_%7BB%7D-v_%7BA%7D%7D)
Since Car B started at the starting line, the distance Car B will be when it passes Car A is
units of distance.