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Grade 10 Mathematics Learners’ Book

INNODEMS

Subsection 2.9.2 Velocity and Acceleration in Different Situations

Activity 2.9.3.

\({\color{black} \textbf{Work in groups}}\)
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  1. Define
    1. Velocity
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    2. Velocity
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  2. A car travels \(200\) meters in \(10\) seconds. What is its average velocity?
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  3. A car accelerates from \(\textbf{75 km/h}\) to \(\textbf{90 km/h}\) in \(\textbf{10 seconds}\)
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    Find it’s acceleration
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  4. A vehicle moving at \(\text{25 m/s}\) applies brakes and comes to a stop in \(\text{5 seconds}\text{.}\)
    1. What is the acceleration of the vehicle?
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    2. Is the acceleration positive or negative? Why?
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\({\color{black} \textbf{Key Takeaway}}\)
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\(\text{Velocity}\) is Speed in a specified direction or the rate of change of displacement with time.
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Symbol of velocity is given as \(\textbf{v}\) while speed is given as \(\textbf{s}\)
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\(v = \frac{\textbf{d}}{\textbf{t}} \)
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where \(\textbf{d}\) represents the distance and \(\textbf{t}\) represents time
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For motion with \(\textbf{constant velocity}\text{,}\) the equation is
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\begin{gather*} \textbf{s = vt} \end{gather*}
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where \(\textbf{v}\) is velocity , \(\textbf{t}\) is time and \(\textbf{s}\) is displacement
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\(\text{Acceleration}\) - The rate of change of velocity with time. SI unit is \(m/s²\)
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Acceleration is given by
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\begin{gather*} \textbf{Acceleration} \, = \frac{\textbf{Change in velocity}}{\textbf{Time Taken}} \end{gather*}
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\begin{gather*} a = \frac{\Delta v}{\Delta t} \end{gather*}
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Negative acceleration is called \(\text{ ( deceleration or retardation)}\)
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For motion with \(\textbf{constant acceleration}\text{,}\) the three equations of motion are:
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  1. \(\text{final velocity}\) = Initial velocity + (acceleration \(\times \) time)
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    \(\text{v = u + at} \)
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    where;
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  2. \(\text{Displacement}\) = \((\text{initial velocity} \times \text{time}) + (\frac{1}{2} \times \text{acceleration} \times \text{time}^2) \)
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    \(s = ut + \frac{1}{2} at^2 \)
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  3. \(\text{Final velocity}^2\) = \(\text{Initial velocity}^2\) + (2 \(\times\) acceleration \(\times\) displacement)
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    \(\textbf{v}^2 = \textbf{u}^2 + \, (2 \times \textbf{acceleration}) \times \textbf{s}^2\)
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Example 2.9.4.

A car starts from rest and accelerates at 2 m/s² for 5 seconds. Find its final velocity.
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Solution.
\(v = u + at \)
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\(= 0 + (2 \times 5) = 10 \textbf{ m/s}\)
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The final velocity is \(10 \textbf{ m/s}\)
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Example 2.9.5.

A rocket accelerates from rest to \(\textbf{250 m/s in 10 seconds}\text{.}\) Calculate the rocket’s acceleration.
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Solution.
Initial velocity \(\textbf{(u) = 0 m/s}\)
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Final velocity \(\textbf{(v) = 250 m/s}\)
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Time (t) = \(\textbf{10 seconds}\)
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Acceleration (a) = (v - u) / t
\begin{gather*} \frac{\textbf{(v - u)}}{\textbf{t}} \end{gather*}
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\begin{gather*} a = \frac{\textbf{(250 m/s - 0 m/s)}}{\textbf{10 seconds}} \end{gather*}
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\begin{gather*} a = \frac{\textbf{(250 m/s)}}{\textbf{10 seconds}} \end{gather*}
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\begin{gather*} \textbf{ a = 25 m/s²} \end{gather*}
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The rocket’s acceleration is \(\textbf{25 m/s²}\)
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Example 2.9.6.

An object decelerates from 20 m/s to 5 m/s in 3 seconds. What is its acceleration?
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Solution.
Initial velocity \(\textbf{(u) = 20 m/s}\)
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Final velocity \(\textbf{(v) = 5 m/s}\)
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Time (t) = \(\textbf{3 seconds}\)
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Acceleration (a) = (v - u) / t
\begin{gather*} \frac{\textbf{(v - u)}}{t} \end{gather*}
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\begin{gather*} a = \frac{\textbf{(5 m/s - 20 m/s)}}{\textbf{3 seconds}} \end{gather*}
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\begin{gather*} a = \frac{\textbf{(-15 m/s)}}{\textbf{3 seconds}} \end{gather*}
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\begin{gather*} \textbf{ a = -5 m/s²} \end{gather*}
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The object’s acceleration is \(\textbf{ -5 m/s²}\text{.}\) The negative sign indicates deceleration (slowing down).
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Exercises Exercises

1.

A car’s velocity changes from \(\textbf{10 m/s to 30 m/s}\) in \(4\) seconds. Find its acceleration.
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2.

A car moves with \(\textbf{8 m/s² }\) acceleration for \(5\) seconds, reaching \(\textbf{40 m/s}\text{.}\) Find its initial velocity.
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3.

A train moving at \(\textbf{ 40 km/h }\) decelerates at \(\textbf{0.5 m/s² }\text{.}\) Find the time taken to stop
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4.

A cyclist’s speed increases from \(\textbf{ 5 m/s to 17 m/s }\) over a period of \(\textbf{ 6 seconds }\) . What is the cyclist’s average acceleration?
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5.

A runner’s velocity changes from \(\textbf{ 3 m/s to 7 m/s }\text{.}\) What is the runner’s average velocity?
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Checkpoint 2.9.7.

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Checkpoint 2.9.8.

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