Question

A bus accelerates uniformly from \[54\,{\text{km}}{{\text{h}}^{{\text{ - 1}}}}\] to \[72\,{\text{km}}{{\text{h}}^{{\text{ - 1}}}}\] in \[10\,{\text{seconds}}\]. Calculate the acceleration in \[{\text{m}}{{\text{s}}^{ - 2}}\].

Answer 1

Hint:First, write down the given quantities. We are asked to find the acceleration in metre per second square so before proceeding for calculation convert the units of the given quantities to the same unit. Here you will need to use first equation of motion and put the given quantities in the equation to find the value of acceleration.

Complete step by step answer:
Given, initial velocity, \[u = 54\,{\text{km}}{{\text{h}}^{{\text{ - 1}}}}\].Final velocity, \[v = 72\,{\text{km}}{{\text{h}}^{{\text{ - 1}}}}\].Time taken, \[t = 10\,{\text{s}}\].We are asked to calculate the acceleration in metre per second square so, we will convert the unit of velocities to metre per second.
\[1\,{\text{hour}} = 3600\,{\text{seconds}}\]
\[\Rightarrow 1\,{\text{km}} = 1000\,{\text{m}}\]
We will use these values to convert kilometre to metre and hour to second for the given velocities.
\[u = 54\,{\text{km}}{{\text{h}}^{{\text{ - 1}}}} \\
\Rightarrow u= 54 \times \dfrac{{1000}}{{3600}}\,{\text{m}}{{\text{s}}^{{\text{ - 1}}}}\]
\[ \Rightarrow u = 15\,{\text{m}}{{\text{s}}^{{\text{ - 1}}}}\]
\[v = 72\,{\text{km}}{{\text{h}}^{{\text{ - 1}}}}
\Rightarrow v = 72 \times \dfrac{{1000}}{{3600}}\,{\text{m}}{{\text{s}}^{{\text{ - 1}}}}\]
\[ \Rightarrow v = 20\,{\text{m}}{{\text{s}}^{{\text{ - 1}}}}\]

From first equation of motion we have,
\[v = u + at\] (i)
where \[u\] is the initial velocity, \[v\] is the final velocity, \[t\] is the time taken and \[a\] is the acceleration.
Putting the values of \[u\], \[v\] and \[t\] we get,
\[20 = 15 + a \times 10\]
\[ \Rightarrow 10a = 20 - 15\]
\[ \Rightarrow 10a = 5\]
\[ \Rightarrow a = \dfrac{5}{{10}}\]
\[ \therefore a = 2\,{\text{m}}{{\text{s}}^{{\text{ - 2}}}}\]

Therefore, the acceleration is \[2\,{\text{m}}{{\text{s}}^{{\text{ - 2}}}}\].

Note: Acceleration can be defined as change in velocity per unit time. We can also solve the above problem by just calculating the difference between the final and the initial velocity and dividing it by the time taken. If the initial and final velocities are the same then the acceleration is zero, that is we can say the body is moving with uniform velocity. And if the final velocity is less than the initial velocity then we can say the acceleration of the body is decreasing or there is retardation.