Question

1. which of the following pairs of graphs shows the distance traveled versus time and the speed versus time for an object uniformly accelerated from rest?

Explanation:

Step1: Recall the equations of motion

For an object uniformly accelerated from rest (\(u = 0\)), the speed - time relation is \(v=u + at\). Since \(u = 0\), we have \(v = at\). This is a linear equation (\(y=mx\) form, where \(y = v\), \(m=a\) (acceleration, constant) and \(x=t\)). So the speed - time graph is a straight line passing through the origin.
The distance - time relation is \(s=ut+\frac{1}{2}at^{2}\). Since \(u = 0\), we have \(s=\frac{1}{2}at^{2}\). This is a quadratic equation (\(y = kx^{2}\) form, where \(y = s\), \(k=\frac{1}{2}a\) (constant) and \(x = t\)). The graph of \(y = kx^{2}\) is a parabola.

Step2: Analyze each option

• Option (A): The speed - time graph is \(v = at\) (correct), but the distance - time graph is not \(s=\frac{1}{2}at^{2}\) (a parabola).
• Option (B): The speed - time graph is not \(v = at\) (not a straight line).
• Option (C): The distance - time graph is not \(s=\frac{1}{2}at^{2}\) (a parabola).
• Option (D): The speed - time graph is \(v = at\) (straight line passing through the origin) and the distance - time graph is \(s=\frac{1}{2}at^{2}\) (a parabola).

Answer:

D.