Introduction to Motion Problems

We have already finished learning how to solve number problems and age problems, so we continue with learning motion problems. Motion problems deal with moving objects such as cars, planes, boats, etc and their speed, distance traveled and time spent traveling. Walking and running are also usually asked in motion problems. Motion problems also include headwind, tailwind, for planes and other flying objects and upstream and downstream problems on boats traveling in rivers.

Average Speed and Actual Speed

In motion problems, you can usually read phrases like the “speed of a car is 60 kilometers per hour.” In many of such problems, it appears that the cars are traveling at a constant speed. In reality, of course, speed is not usually constant. In the course of the travel, a car will accelerate, decelerate, or even stop at an intersection if the light is red. This means that when we talk about “60 kilometers per hour” in moition problems, in reality, they mean average speed.  That is, in one hour, a car usually travels 60 kilometers given normal traffic conditions. Usually, in math problems, rate and speed means the same thing.

Distance = Rate Times Time

A car that travels 70 kilometers per hour can travel 140 kilometers in two hours. We can write this as an equation shown below

140km = (70 km/hr)(2hrs).

Since 140 kilometers is the distance (d), 70 km/hr is the rate (r), and 2 hours is the time (t), it follows that

distance = rate $\times$ time

or $d = rt$.

This is a basic formula and you don’t really need to memorize it if you understand the concept. Now, for the examples, let’s solve two basic  motion problems.

Problem 1

Aron goes to a convenient store every week using a bicycle. If his average speed using a bicycle is 30 km/hr, how far is the convenient store if it takes him 15 minutes to go there?

Solution 1

This problem can be solved by inspection, but we will setup the equation in solution 2. For inspection, 15 minutes is 1/4 of an hour, so if it takes Aron to travel 30 km per hour, then he will travel 1/4 of it given 1/4 of an hour. Therefore, 1/4 of 30 kilometers is 7.5 kilometers.

Solution 2

The second solution is needed so you would learn how to set up equations. Later problems will be a lot harder than this, so you will need to set up the equations correctly.

distance = ?

rate = 30 km/h

time 1/4 hr

$d = rt$

$d = 30 (\frac{1}{4}) = 7.5$ km.

Problem 2

It took 4.5 hours from Ninoy Aquino International Airport (Philippines) to Narita Airport (Japan).  If the route of the plane is around 2250 kilometers, what is its speed?

Solution

$t = 4.5$ hours

$r = ?$

$d = 2250$ kilometers

From the original equation, we have

$d = rt$

$2250 = 4.5(r)$

To get $r$, divide both sides of the equation by $4.5$ giving us

$r = 500$.

So, the airplane is traveling 500 kilometers per hour.

That’s it! In the next post, we discuss how to solve more motion problems.