Question

A three-wheeler starts from rest, accelerates uniformly with \[1m/{s^2}\] on a straight road for \[10s\] , and then moves with uniform velocity. Plot the distance covered by the vehicle during the nth second \[\left( {n = 1,2,3,..} \right)\] versus \[n\] . What do you expect this plot to be during accelerated motion: a straight line or a parabola?

Answer 1

Hint: We are asked to plot the distance time graph of the given motion and see the shape of the curve. We start by writing down the data and finding the value for the distance travelled in \[n\] seconds. This gives us an idea about how distance varies with the time and hence we can find the trend of the graph.

Formulas used:
The distance travelled by the three-wheeler in \[n\] seconds is given by the formula,
\[S = u + \left( {2n - 1} \right)\dfrac{a}{2}\]
Where \[u\] is the initial velocity of the three-wheeler and \[a\] is the acceleration of the three-wheeler.

Complete step by step answer:
Let us start by noting down the data given in the question. As the body starts from rest, The initial velocity of the body will be \[u = 0\]. The acceleration of the body is said to be uniform for a time of ten seconds and this acceleration is given as, \[a = 1m/{s^2}\].

Now that we have all the values, we find the distance travelled in \[n\] seconds by the three-wheeler using the formula,
\[S = u + \left( {2n - 1} \right)\dfrac{a}{2}\]
Substituting the values we have, we get
\[S = u + \left( {2n - 1} \right)\dfrac{a}{2} \\
\Rightarrow S = 0 + \left( {2n - 1} \right)\dfrac{1}{2} \\
\Rightarrow S = n - \dfrac{1}{2}\]
The above equation means that the distance travelled by the three-wheeler in seconds will be proportional to the distance. That is, the quantities in the x and y axes are proportional to each other.

Whenever there is a linear direct proportion relation in between any two quantities, the graph of the two quantities is always a straight line because the equation of a straight line in general form in slope intercept form is also a direct relation in x and y as $y = mx + c$. Hence, the graph of distance covered in nth seconds versus $n$ will be a straight line.

Note: A distance-time graph is one of the useful ways in representing the motion of an object. It shows how the distance moved by a body is dependent on the time taken for the travel. The distance travelled is plotted on the vertical (y) axis and the time taken is plotted on the horizontal (x) axis.