## How to Solve Motion Problems Part 1

Now that I have introduced to you motion problems, let us solve the type of problems that usually appear in textbooks as well as examinations. In this part of the series, we will learn how to analyze and solve a problem involving objects moving in the same direction. This is our third problem in the How to Solve Motion Problems Series.

Problem 3

Car A left Math City going to English City at an average speed of 40 kilometers per hour.  Two hours later, Car B traveling 60 kilometers per hour leaves the same place for English City. In how many hours, will the car B overtake Car A?

Solution 1

This problem can be solved manually by creating the table as shown in the next figure. In the first column, we have the cars, and in the first row the number of hours. Car A traveled 40 kilometers and Car B travels 60 kilometers per hour. Since Car B only started traveling after two hours, so, after three hours, Car A has traveled 120 kilometers and Car B just 60 kilometers. Now, since Car B is faster, it will eventually overtake Car A which is after 6 hours shown below.  Having the same distance traveled in this problem means that Car B overtakes Car A since they are traveling the same route.

Now notice that the question in “how many hours” means that the counting starts when Car B left. Therefore, the answer is 4 hours because in 4 hours, car B traveled 240 kilometers, the same distance traveled by Car A.

Solution 2

In solving Motion problems, it is always helpful to create a table of the given distance, rate, and time. The rate r of Car A is 40km/hr and Car B is 60 km/hr. Now, for the time, Car A started two hours earlier, so if, for example, it has traveled 5 hours, Car B has only traveled 3 hours. It means that the time Car A traveled is 2 hours more than that of Car B. This means that if Car B traveled $x$ hours, then car A traveled $x + 2$ hours. Now, as we have discussed in the previous post, $d = rt$, so for column $d$ in the table below, we just multiply the rate $(r)$ and the time $(t)$. Now what to do next?

At the exact time Car B overtakes Car A, which is what is asked above, their distance traveled will be equal. This means that

distance traveled by Car A = distance traveled by Car B

or using the expressions above

$40(x + 2) = 60x$

Note that the $x$ in this equation is the time from the table above, so the answer will be in hours. We solve the equation by simplifying the left hand side first by distributive property

$40x + 80 = 60x$.

Subtracting $40x$ from both sides results to

$80 = 20x$ which simplifies to

$4 = x$.

This means that Car A overtakes car B in 4 hours after Car B left. This confirms the answer in Solution 1.

This post is a bit detailed in order to let you understand how to solve the equations. The next two problems will be shorter.