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Grade 10 Mathematics Learners’ Book

INNODEMS

Subsection 2.9.1 Speed

Activity 2.9.1.

\({\color{black} \textbf{Work in groups}}\)
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\(\textbf{Materials :}\) Stop watch, Measuring tape and ball/round object
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  1. Roll a ball or any round object on the table
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  2. Measure the rolling distance using the tape measure
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  3. Record the time when rolling
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  4. Find the speed of the rolling ball
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Activity 2.9.2.

\({\color{black} \textbf{Work in groups}}\)
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A family drives 120 km at 60 km/h, stops for a 15-minute break, and then drives another 80 km at 40 km/h. What is the average speed for the entire trip?
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  1. Calculate the time for each driving segment
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  2. Convert the break time to hours
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  3. Calculate the total distance
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  4. Calculate the total time
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  5. Calculate the average speed
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\({\color{black} \textbf{Key Takeaway}}\)
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\(\text{Speed}\) is the rate of change of distance with time. SI unit: \(km/h\) .
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Speed normally varies over time, so the average speed is often used
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\begin{gather*} \textbf{Average speed} = \frac{\textbf{Total distance covered}}{\textbf{Total time taken}} \end{gather*}
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Example 2.9.1.

A car travels 100 km in 2 hours. Find its speed.
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Solution.
\(\textbf{Speed} = \frac{\textbf{distance}}{\textbf{time}}\)
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\(\textbf{Speed} = \frac{100}{2} = 50 \text{ km/h} \)
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Example 2.9.2.

A motorist covers a distance of 360 km in 3 hours travelling from busia to bahangongo. Calculate the average speed.
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Solution.
\(\textbf{Speed} = \frac{\textbf{distance}}{\textbf{time}}\)
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\begin{gather*} \textbf{Average speed} = \frac{\textbf{360 km}}{\textbf{3 h}} \end{gather*}
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\begin{gather*} \textbf{120 km/h} \end{gather*}
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Example 2.9.3.

Alex is cycling from home to the market. He rides the first 8 km at a speed of 16 km/h. After reaching a park, Alex takes a 20-minute break to play football with his friends. Then, he cycles the remaining 6 km to the market at a speed of 12 km/h. What was Alex’s average speed for the entire journey?
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Solution.
  1. Calculating the time for each cycling part.
    • \(\textbf{part 1}\)
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      Distance: \(\textbf{8 km}\)
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      Speed: \(\textbf{16 km/h}\)
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      \begin{gather*} \textbf{Time}\, = \, \frac{\textbf{Distance}}{\textbf{Speed}} \, = \, \frac{\textbf{8 km}}{\textbf{16 km/h}} \, = \textbf{0.5 hours} \end{gather*}
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    • \(\textbf{part 2}\)
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      Distance: \(\textbf{6 km}\)
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      Speed: \(\textbf{12 km/h}\)
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      \begin{gather*} \textbf{Time}\, = \, \frac{\textbf{Distance}}{\textbf{Speed}} \, = \, \frac{\textbf{6 km}}{\textbf{12 km/h}} \, = \textbf{0.5 hours} \end{gather*}
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  2. Convert the break time to hours
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    \(\frac{\textbf{20 minutes}}{\textbf{60 minutes}} = \frac{1}{3} \, \text{hours}\)
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  3. Calculating the total distance gives
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    \begin{gather*} \textbf {Total distance = 8 km + 6 km = 14 km } \end{gather*}
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  4. Calculating the total time gives
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    Total time =
    \begin{gather*} \frac{1}{2} \, \text{hours} + \frac{1}{3} \, \text{hours} + \frac{1}{2} \, \text{hours} = 1\frac{1}{3} \, \text{hours} \end{gather*}
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  5. Now Calculating the average speed gives
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    \(\textbf{Average speed} \, = \, \frac{\textbf{Total distance}}{\textbf{Total time}}\)
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    \(\textbf{Average speed} \, = \, \frac{\textbf{14 km}}{\textbf{1.33 hours}} \, = \, \textbf{10.53 km/h} \)
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Alex’s average speed for the entire journey was approximately \(\textbf{10.53 km/h} \text{.}\)
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Exercises Exercises

1.

Juma cycled for \(3\) hours to a trading centre \(60\) km away, then drove \(250 \,\)km in a van at \(50 \,\) km/h and finally cycled home at \(20 \,\) km/h. Find the average speed for the whole journey.
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2.

Sarah is rushing to school. She walks the first \(\textbf{500 meters} \,\) at \(\, \textbf{5 km/h}\text{,}\) realizing she’s late, she then sprints the next \(\textbf{300 meters} \,\) at \(\, \textbf{10 km/h}\) What was her average speed for the entire trip to school?
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3.

John walks to the store, a distance of \(\textbf{1 km}\text{,}\) at a speed of \(\textbf{4 km/hr}\text{.}\) He spends \(\textbf{15 minutes}\) at the store, and then walks back at \(\textbf{5 km/hr}\text{.}\) What was John’s average speed for the entire trip, including the time spent at the store
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4.

A hiker covered \(\textbf{ 12 km in 3 hours}\text{.}\) After taking a \(\textbf{15 minute}\) break, what speed must the hiker maintain to reach their destination within a total travel time of \(\textbf{4 hours}\text{?}\)
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5.

A runner completed \(\textbf{ 10 km in 1.5 hours}\text{.}\) After a \(\textbf{5 minute}\) rest, what pace does the runner need to maintain to finish a total distance within \(\textbf{2 hours}\text{?}\)
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6.

A train journey consisted of three segments. The train traveled for \(\textbf{ 3 hours at 90 km/h}\text{,}\) then paused form \(\textbf{ 0.75 hours}\) at a station, and finally continued for \(\textbf{ 2 hours at 70 km/h}\text{.}\) What was the average speed of the train for the entire journey?
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