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 Da beyond the starting line at t=0. The starting line is at x=0. Car A travels at a constant speed Va. Car B starts at the starting line but has a better engine than Car A, and thus Car B travels at a constant speed Vb, which is greater than Va.
A. How long after Car B started the race will Car B catch up with Car A?
B. How far from Car B's starting line will the cars be when Car B passes Car A?
To find the time it takes for Car B to catch up with Car A, we need to analyze their relative motion.
Let's assume that Car B catches up with Car A at time t. At that time, both cars will have traveled the same distance.
Since Car A has a head start, the distance it travels at time t is (Da + Va*t).
Car B starts at the starting line, so the distance it travels at time t is Vb*t.
To find when Car B catches up with Car A, we equate the distances traveled:
Da + Va*t = Vb*t
Now, we can solve for t:
Da = Vb*t - Va*t
Da = (Vb-Va)*t
Dividing both sides by (Vb-Va), we get:
t = Da / (Vb - Va)
Therefore, the time it takes for Car B to catch up with Car A is t = Da / (Vb - Va).
To find the distance from Car B's starting line when Car B passes Car A, we can substitute the value of t back into either Car A's or Car B's distance equation.
Using Car B's distance equation:
Distance from Car B's starting line = Vb*t
Substituting the value of t we found earlier, we have:
Distance from Car B's starting line = Vb * (Da / (Vb - Va))
Simplifying, we get:
Distance from Car B's starting line = (Vb * Da) / (Vb - Va)
Therefore, the cars will be (Vb * Da) / (Vb - Va) distance away from Car B's starting line when Car B passes Car A.
To find the time it takes for Car B to catch up with Car A, we can set up an equation based on their relative speeds.
Let's assume that Car B catches up with Car A at time t.
At time t, Car B has traveled a distance equal to Car A's head start plus the distance that Car A has traveled.
The distance traveled by Car A can be calculated as Da + Va * t.
The distance traveled by Car B can be calculated as Vb * t.
Since Car B catches up with Car A when they have traveled the same distance, we can set up the equation:
Da + Va * t = Vb * t
To solve for t, we can rearrange the equation:
Da = Vb * t - Va * t
Da = (Vb - Va) * t
Now we can solve for t:
t = Da / (Vb - Va)
Therefore, to find the time it takes for Car B to catch up with Car A, divide the head start distance (Da) by the difference in their speeds (Vb - Va).
To find the distance from Car B's starting line where the cars will be when Car B passes Car A, substitute the value of t into either Car A's or Car B's distance equation:
Distance from Car B's starting line = Vb * t
Now, you can plug in the values of Da, Va, and Vb, and calculate the answers to parts A and B of the question.
When car B catches up, its' distance
traveled will be greater than car A by
an amount equal to Da; but their time
on the road will be equal.
A. da = Va*t = distance car A traveled.
Da + da = Vb*t = distance traveled by car B,
t = (Da + da) / Vb.
B. Da + da = distance from car B start-
ing line.