5.3.1: Position, Speed, and Direction

Lesson

Let's use signed numbers to represent movement.

Exercise \(\PageIndex{1}\): Distance, Rate, Time

1. An airplane moves at a constant speed of 120 miles per hour for 3 hours. How far does it go?
2. A train moves at constant speed and travels 6 miles in 4 minutes. What is its speed in miles per minute?
3. A car moves at a constant speed of 50 miles per hour. How long does it take the car to go 200 miles?

Exercise \(\PageIndex{2}\): Going Left, Going Right

Figure \(\PageIndex{1}\)

1. After each move, record your location in the table. Then write an expression to represent the ending position that uses the starting position, the speed, and the time. The first row is done for you.

   Table \(\PageIndex{1}\)

   starting position	direction	speed (units per second)	time (seconds)	ending position (units)	expression
   \(0\)	right	\(5\)	\(3\)	\(+15\)	\(0+5\cdot 3\)
   \(0\)	left	\(4\)	\(6\)
   \(0\)	right	\(2\)	\(8\)
   \(0\)	right	\(6\)	\(2\)
   \(0\)	left	\(1.1\)	\(5\)

2. How can you see the direction of movement in the expression?
3. Using a starting position \(p\), a speed \(s\), and a time \(t\), write two expressions for an ending position. One expression should show the result of moving right, and one expression should show the result of moving left.

Exercise \(\PageIndex{3}\): Velocity

A traffic safety engineer was studying travel patterns along a highway. She set up a camera and recorded the speed and direction of cars and trucks that passed by the camera. Positions to the east of the camera are positive, and to the west are negative.

Figure \(\PageIndex{2}\): Number line. To the left is the word west. To the right is the word east. Three evenly spaced tick marks labeled, from left to right, negative 100, 0, plus 100.

Vehicles that are traveling towards the east have a positive velocity, and vehicles that are traveling towards the west have a negative velocity.

1. Complete the table with the position of each vehicle if the vehicle is traveling at a constant speed for the indicated time period. Then write an equation.

   Table \(\PageIndex{2}\)

   velocity (meters per second)	time after passing the camera (seconds)	ending position (meters)	equation describing the position
   \(+25\)	\(+10\)	\(+250\)	\(25\cdot 10=250\)
   \(-20\)	\(+30\)
   \(+32\)	\(+40\)
   \(-35\)	\(+20\)
   \(+28\)	\(0\)

2. If a car is traveling east when it passes the camera, will its position be positive or negative 60 seconds after it passes the camera? If we multiply two positive numbers, is the result positive or negative?
3. If a car is traveling west when it passes the camera, will its position be positive or negative 60 seconds after it passes the camera? If we multiply a negative and a positive number, is the result positive or negative?

Are you ready for more?

In many contexts we can interpret negative rates as "rates in the opposite direction." For example, a car that is traveling -35 miles per hour is traveling in the opposite direction of a car that is traveling 40 miles per hour.

1. What could it mean if we say that water is flowing at a rate of -5 gallons per minute?
2. Make up another situation with a negative rate, and explain what it could mean.

Summary

We can use signed numbers to represent the position of an object along a line. We pick a point to be the reference point, and call it zero. Positions to the right of zero are positive. Positions to the left of zero are negative.

Figure \(\PageIndex{3}\): A number line with the numbers negative 10 through 10 indicated. A point is indicated at zero and is labeled reference point. Another point is indicated at negative 4 and is labeled 4 units to the left of zero. A third point is indicated at 7 and is labeled 7 units to the right of zero.

When we combine speed with direction indicated by the sign of the number, it is called velocity. For example, if you are moving 5 meters per second to the right, then your velocity is +5 meters per second. If you are moving 5 meters per second to the left, then your velocity is -5 meters per second.

If you start at zero and move 5 meters per second for 10 seconds, you will be \(5\cdot 10=50\) meters to the right of zero. In other words, \(5\cdot 10=50\).

If you start at zero and move -5 meters per second for 10 seconds, you will be \(5\cdot 10=50\) meters to the left of zero. In other words,

\(-5\cdot 10=-50\)

In general, a negative number times a positive number is a negative number.

Practice

Exercise \(\PageIndex{4}\)

A number line can represent positions that are north and south of a truck stop on a highway. Decide whether you want positive positions to be north or south of the truck stop. Then plot the following positions on a number line.

1. The truck stop
2. \(5\) miles north of the truck stop
3. \(3.5\) miles south of the truck stop

Exercise \(\PageIndex{5}\)

1. How could you distinguish between traveling west at 5 miles per hour and traveling east at 5 miles per hour without using the words “east” and “west”?
2. Four people are cycling. They each start at the same point. (0 represents their starting point.) Plot their finish points after five seconds of cycling on a number line.
   • Lin cycles at 5 meters per second
   • Diego cycles at -4 meters per second
   • Elena cycles at 3 meters per second
   • Noah cycles at -6 meters per second

Exercise \(\PageIndex{6}\)

Find the value of each expression.

1. \(16.2+-8.4\)
2. \(\frac{2}{5}-\frac{3}{5}\)
3. \(-9.2+-7\)
4. \(-4\frac{3}{8}-(-1\frac{1}{4})\)

(From Unit 5.2.5)

Exercise \(\PageIndex{7}\)

For each equation, write two more equations using the same numbers that express the same relationship in a different way.

1. \(3+2=5\)
2. \(7.1+3.4=10.5\)
3. \(15-8=7\)
4. \(\frac{3}{2}+\frac{9}{5}=\frac{33}{10}\)

(From Unit 5.2.4)

Exercise \(\PageIndex{8}\)

A shopper bought a watermelon, a pack of napkins, and some paper plates. In his state, there is no tax on food. The tax rate on non-food items is 5%. The total for the three items he bought was $8.25 before tax, and he paid $0.19 in tax. How much did the watermelon cost?

(From Unit 4.3.1)

Exercise \(\PageIndex{9}\)

Which graphs could not represent a proportional relationship? Explain how you decided.

Figure \(\PageIndex{4}\)

(From Unit 2.4.1)