148. Speed, Distance and Time

Learning Intentions

• To understand that speed is a rate relating distance and time
• find an average speed (given a distance and the time taken)
• find the distance travelled (given an average speed and the time taken)
• find the time taken (given an average speed and the distance)
• convert between different units of speed

Pre-requisite Summary

• Know that a rate compares two quantities measured in different units
• Understand that distance is measured in units such as m and km, and time in units such as s, min and h
• Be able to convert between units of time
• Be able to convert between units of length
• Be able to multiply and divide whole numbers and decimals accurately
• Understand that average means total amount divided by total number of equal units
• Know that “per” means “for each” or “for one”

Worked Examples

Worked Example 1

State what each speed means:
a) $60\text{ km/h}$
b) $5\text{ m/s}$
c) $12\text{ km/h}$

Worked Example 2

Find the average speed:
a) $150$ km travelled in $3$ h
b) $240$ m travelled in $40$ s
c) $18$ km travelled in $1.5$ h

Worked Example 3

Find the distance travelled:
a) speed $=80\text{ km/h}$, time $=4$ h
b) speed $=6\text{ m/s}$, time $=12$ s
c) speed $=45\text{ km/h}$, time $=2.5$ h

Worked Example 4

Find the time taken:
a) distance $=180$ km, speed $=60\text{ km/h}$
b) distance $=300$ m, speed $=5\text{ m/s}$
c) distance $=90$ km, speed $=45\text{ km/h}$

Worked Example 5

Convert between units of speed:
a) $72\text{ km/h}$ to $\text{m/s}$
b) $10\text{ m/s}$ to $\text{km/h}$
c) $54\text{ km/h}$ to $\text{m/s}$

Worked Example 6

Solve each worded problem:
a) A car travels $210$ km in $3$ h. Find its average speed.
b) A cyclist rides at $24\text{ km/h}$ for $2.5$ h. Find the distance travelled.
c) A runner travels $800$ m at an average speed of $4\text{ m/s}$. Find the time taken.

Problems

Problem 1

State what each speed means:
a) $90\text{ km/h}$
b) $8\text{ m/s}$
c) $15\text{ km/h}$

Problem 2

Find the average speed:
a) $200$ km travelled in $4$ h
b) $360$ m travelled in $60$ s
c) $27$ km travelled in $1.5$ h

Problem 3

Find the distance travelled:
a) speed $=70\text{ km/h}$, time $=5$ h
b) speed $=4\text{ m/s}$, time $=20$ s
c) speed $=36\text{ km/h}$, time $=1.5$ h

Problem 4

Find the time taken:
a) distance $=240$ km, speed $=80\text{ km/h}$
b) distance $=500$ m, speed $=10\text{ m/s}$
c) distance $=72$ km, speed $=36\text{ km/h}$

Problem 5

Convert between units of speed:
a) $36\text{ km/h}$ to $\text{m/s}$
b) $15\text{ m/s}$ to $\text{km/h}$
c) $90\text{ km/h}$ to $\text{m/s}$

Problem 6

Solve each worded problem:
a) A bus travels $280$ km in $4$ h. Find its average speed.
b) A swimmer moves at $2\text{ m/s}$ for $45$ s. Find the distance travelled.
c) A train travels $360$ km at an average speed of $90\text{ km/h}$. Find the time taken.

Exercises

Understanding and Fluency

1. Complete each statement:
   a) Speed is a ______ relating distance and time
   b) Average speed is found by dividing ______ by ______
   c) Distance is found by multiplying ______ by ______
   d) Time is found by dividing ______ by ______

2. State what each speed means:
   a) $50\text{ km/h}$
   b) $3\text{ m/s}$
   c) $100\text{ km/h}$
   d) $12\text{ m/s}$

3. Find the average speed:
   a) $120$ km in $2$ h
   b) $300$ km in $5$ h
   c) $150$ m in $30$ s

4. Find the average speed:
   a) $480$ m in $80$ s
   b) $54$ km in $3$ h
   c) $22.5$ km in $1.5$ h

5. Find the distance travelled:
   a) $60\text{ km/h}$ for $4$ h
   b) $8\text{ m/s}$ for $25$ s
   c) $72\text{ km/h}$ for $2.5$ h

6. Find the time taken:
   a) $150$ km at $50\text{ km/h}$
   b) $420$ m at $7\text{ m/s}$
   c) $96$ km at $48\text{ km/h}$

7. Convert between units of speed:
   a) $18\text{ km/h}$ to $\text{m/s}$
   b) $20\text{ m/s}$ to $\text{km/h}$
   c) $108\text{ km/h}$ to $\text{m/s}$
   d) $25\text{ m/s}$ to $\text{km/h}$

8. Solve each:
   a) A car travels at $90\text{ km/h}$ for $3$ h. Find the distance.
   b) A bike travels $45$ km in $2$ h. Find the average speed.
   c) A runner travels $1000$ m at $5\text{ m/s}$. Find the time taken.

Reasoning

9. Explain why speed is a rate and not just a single measurement.

10. A student says that average speed is found by adding the distance and the time. Explain the mistake.

11. Noah says that if a car travels at $60\text{ km/h}$ for $2$ h, then the distance is $62$ km. Is he correct? Explain.

12. Explain why the units are important when writing an answer for speed.

13. A student says that $72\text{ km/h}=72\text{ m/s}$. Describe the error.

Problem-solving

14. A train travels $360$ km in $4$ h. Find its average speed.

15. A boat moves at $18\text{ km/h}$ for $3.5$ h. Find the distance travelled.

16. A runner travels $1500$ m at an average speed of $5\text{ m/s}$. Find the time taken.

17. A car travels at $108\text{ km/h}$. Convert this speed to $\text{m/s}$.

18. A cyclist travels $42$ km in $1.5$ h. Find the average speed.

19. A bus travels at $80\text{ km/h}$ for $2.25$ h. Find the distance travelled.

Potential Misunderstandings

• Students may confuse speed with distance or time
• Students may use the wrong formula, such as dividing time by distance
• Students may add distance and time instead of dividing or multiplying
• Students may forget to include units such as km/h or m/s in the answer
• Students may mix kilometres with seconds or metres with hours without converting
• Students may think average speed means the speed at one instant rather than over the whole trip
• Students may make errors converting between km/h and m/s
• Students may divide when they should multiply when finding distance
• Students may multiply when they should divide when finding time