Motion problems

Algebra, grade 8 Lesson 17 of 27Learning

Motion problems

Distance, speed and time

The units go together: km/h, hours and km. Half an hour is

\displaystyle 0.5

hours, and 15 minutes are

\displaystyle 0.25

hours.

Moving at a constant speed, you cover the same distance every hour. So the distance is the speed times the time:

\displaystyle s

\displaystyle =

\displaystyle v\cdot t

$\displaystyle s$ – the distance (in km), $\displaystyle v$ – the speed (in km/h), $\displaystyle t$ – the time (in hours).
At 60 km/h you cover 60 km every hour, and in two hours $\displaystyle {60\cdot2}={120}$ km.
From the same formula: the time is the distance divided by the speed. 120 km at 80 km/h take $\displaystyle {\frac{120}{80}}={1.5}$ hours.
And the speed is the distance divided by the time.

Exercise 1

Calculate.

A train travels for 3.5 hours at 90 km/h.
What distance does it cover?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

A cyclist rides 45 km at 18 km/h.
How many hours does she ride?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

A 6 km walk took an hour and a half.
What was the speed?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Example: toward each other

The truck left an hour after the car, so it drove one hour less:

\displaystyle x-1

The distance between two cities is 360 km.
A car leaves one city at 80 km/h.
An hour later a truck leaves the other city toward it, at 60 km/h.
After how many hours do they meet?

$\displaystyle x$ – the car’s driving time, in hours. A table puts the two motions in order: a row for each, and in each row the distance is the speed times the time.

	Speed (km/h)	Time (hours)	Distance (km)
Car	80	$\displaystyle x$	$\displaystyle 80x$
Truck	60	$\displaystyle x-1$	$\displaystyle 60(x-1)$

When they meet, together they have covered the whole distance between the cities:

\begin{aligned}80x+60(x-1)&=360\\ 140x-60&=360 &&/+60\\ 140x&=420 &&/:140\\ x&=3\end{aligned}

Answer: they meet 3 hours after the car left. Check: the car covered $\displaystyle {80\cdot3}={240}$ km, and the truck $\displaystyle {60\cdot2}={120}$ km – together 360 km.

Exercise 2

The distance between two villages is 84 km.
A cyclist leaves one village at 18 km/h.
An hour later a cyclist leaves the other village toward him, at 15 km/h.
How many hours after the first cyclist left will they meet?
How far from his village will the first cyclist be then?
Make a table as in the example.

After how many hours

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

The first rider from his village (km)

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Example: one after the other

Dana sets off on a walk at 6 km/h.
Two hours later Ron sets off along the same path, on a bike, at 18 km/h.
After how many hours does Ron catch up with Dana?

$\displaystyle x$ – Ron’s riding time; Dana walks two hours more, $\displaystyle x+2$ .

	Speed (km/h)	Time (hours)	Distance (km)
Ron	18	$\displaystyle x$	$\displaystyle 18x$
Dana	6	$\displaystyle x+2$	$\displaystyle 6(x+2)$

When Ron catches up with Dana, they are at the same point – they have covered the same distance:

\begin{aligned}18x&=6(x+2)\\ 18x&=6x+12 &&/-6x\\ 12x&=12 &&/:12\\ x&=1\end{aligned}

Answer: after an hour of riding, 18 km from the start. Check: Dana walked for 3 hours, $\displaystyle {6\cdot3}={18}$ km.

Exercise 3

A bus leaves a station at 8:00 and travels at 60 km/h.
A car leaves the same station at 9:00, along the same road, at 90 km/h.
At what time will the car catch up with the bus?
How far from the station will that be?

The time

The distance from the station (km)

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Example: there and back

The distance there and the distance back are equal – only the times differ.

A cyclist rode to a lake at 20 km/h, and came back the same way at 15 km/h.
The ride there and back took 7 hours.
How far away is the lake?

$\displaystyle x$ – the riding time there; the time back is $\displaystyle 7-x$ . The distance there, $\displaystyle 20x$ , equals the distance back, $\displaystyle 15(7-x)$ :

\begin{aligned}20x&=15(7-x)\\ 20x&=105-15x &&/+15x\\ 35x&=105 &&/:35\\ x&=3\end{aligned}

Answer: the distance there is $\displaystyle {20\cdot3}={60}$ km – the lake is 60 km away. Check: back, $\displaystyle {15\cdot4}={60}$ km.

You can also let $\displaystyle x$ be the distance. Then the times are $\displaystyle \frac{x}{20}$ and $\displaystyle \frac{x}{15}$ , and the equation is $\displaystyle {\frac{x}{20}+\frac{x}{15}}={7}$ . Multiplying by 60 gives $\displaystyle {3x+4x}={420}$ , and again $\displaystyle {x}={60}$ .

Exercise 4

Noa walked up a path to the top of a hill at 3 km/h, and down the same path at 5 km/h.
Going up and down took 4 hours together.
How long is the path?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 5

A train at 120 km/h covers the road between two cities in one hour less than a bus at 80 km/h.
How long is the road between the cities?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 6

Two cars leave the same point at the same time, in opposite directions.
One travels at 70 km/h, the other at 50 km/h.
After how long will they be 300 km apart?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: algebra, grade 8

Open as a page

The linear function

A linear function – $\displaystyle m$ the slope; the line crosses the $\displaystyle y$ -axis at $\displaystyle (0,b)$
$\displaystyle y=mx+b$
The slope of the line through $\displaystyle (x_1,y_1)$ and $\displaystyle (x_2,y_2)$
$\displaystyle m=\frac{y_2-y_1}{x_2-x_1}$
Parallel lines have the same slope
$\displaystyle m_1=m_2$
A line through the origin – $\displaystyle {b}={0}$
$\displaystyle y=mx$
A horizontal line – the slope is 0
$\displaystyle y=b$

Word problems

$\displaystyle p$ percent of $\displaystyle x$
$\displaystyle \frac{p}{100}\cdot x$
$\displaystyle s$ the distance, $\displaystyle v$ the speed, $\displaystyle t$ the time
$\displaystyle s$ $\displaystyle =$ $\displaystyle v\cdot t$

Inequalities

Adding or subtracting the same number on both sides – the sign stays
$\displaystyle a+c<b+c$ $\displaystyle a<b$
Multiplying or dividing by a positive number – the sign stays
$\displaystyle a\cdot c<b\cdot c$ $\displaystyle a<b,\ c>0$
Multiplying or dividing by a negative number – the sign turns round
$\displaystyle a\cdot c>b\cdot c$ $\displaystyle a<b,\ c<0$