Exercise 4.3
A ball rolls down an inclined track as shown in Fig. 4.6. Is its motion, a straight line motion? Assuming the starting point of the ball (O) to be the origin, can its motion from O to D be depicted using a horizontal line as shown in Fig. ? Are the values of total distance travelled and magnitude of displacement from O equal or different at positions A, B, C and D?
Check this one against your book
NCERT’s printed answer for this exercise does not match its own question. This working follows the question as printed.
Yes, it is motion in a straight line. The sloping face of the track in Fig. $\displaystyle 4.6$ is a single straight segment running from O down to D; the ball stays on it throughout. A straight line need not be horizontal — only the path matters, not its tilt.
Yes, it can be depicted on a line like Fig. 4.3. Take O as the origin and mark the distances measured along the track. Fig. $\displaystyle 4.6$ dimensions the slope in four pieces, \(\displaystyle \text{OA} = 40\ \text{cm} \), \(\displaystyle \text{AB} = 10\ \text{cm} \), \(\displaystyle \text{BC} = 20\ \text{cm} \), \(\displaystyle \text{CD} = 30\ \text{cm} \), so on such a line A is at \(\displaystyle 40\ \text{cm} \), B at \(\displaystyle 40 + 10 = 50\ \text{cm} \), C at \(\displaystyle 50 + 20 = 70\ \text{cm} \) and D at \(\displaystyle 70 + 30 = 100\ \text{cm} \). The line records only position along the path with respect to the origin; the tilt of the real track adds no information once the motion is along one straight line.
The two are equal at every one of A, B, C and D. The ball rolls one way and never turns back, so by the chapter's Note — for motion in a straight line the total distance travelled and the magnitude of displacement are equal if the object moves in one direction — the total distance travelled from O equals the magnitude of displacement from O: \(\displaystyle 40\ \text{cm} \) at A, \(\displaystyle 50\ \text{cm} \) at B, \(\displaystyle 70\ \text{cm} \) at C and \(\displaystyle 100\ \text{cm} \) at D, the displacement being directed from O down the slope towards D.
What does differ is the value from one position to the next, \(\displaystyle 40 \rightarrow 50 \rightarrow 70 \rightarrow 100\ \text{cm} \); at each single position, distance and magnitude of displacement agree.