EN हिं

NCERT Solutions for Class 9th Maths Chapter 6 Chapter opening: the 4 × 100 m relay track — In-text Questions

Book page 118 Updated on2026-09-08

Q1.
Those in the outer lanes seem to be starting ahead of those in the inner lanes while the finish line is the same for all of them. What could be the reason for this?
Answer

Because the lanes are not the same length. The finish line is common, so if everyone also started on a common line the outer runners would have to cover more ground.

Why it happens: a running track is two straight stretches joined by two semicircular bends. On the straights every lane is exactly as long as every other. On the bends, lane 2 curves round a semicircle of larger radius than lane 1, lane 3 larger than lane 2, and so on. Since the length of a semicircle is πr, a bigger r means a longer path. The starting points are therefore moved forward — staggered — by exactly the extra distance the outer lane would otherwise run.
Extra length in one lane out
= 2 semicircles of radius (r + w) − 2 semicircles of radius r
= 2π(r + w) − 2πr
= 2πw, where w is the lane width

Notice what dropped out: the answer does not contain r at all. The stagger depends only on the width of a lane.

Q2.
Do you think the stagger gives anyone (those in the outer lanes or in the inner lanes) an unfair advantage? Why or why not?
Answer

No. The stagger is calculated so that every athlete runs exactly 400 m (or exactly 100 m in a relay leg). It removes an unfairness rather than creating one.

Why it happens: the organisers set the stagger equal to the extra curved distance, 2π × (lane width) per full lap. After that adjustment, the total path length is identical in every lane. What still differs is the tightness of the bend: an inner-lane runner turns through a sharper curve and has to lean more, while an outer-lane runner runs a gentler curve but cannot see the runners behind her. Athletes and coaches argue about which of these matters more, but the distance is exactly equal.
Did you know? In races run entirely in lanes (100 m, 200 m, 400 m, and the 4 × 100 m relay) the stagger is used. In longer races the runners break for the inside lane after the first bend, so a different, smaller stagger is used.
Q3.
On what basis can the organisers work out the length of the stagger between lanes?
Answer

On the formula for the length of a circle. That is the whole reason the chapter starts here.

Length of a lap in a lane of radius r
= 2 × (straight) + circumference of the circle of radius r
= 2s + 2πr

Stagger between two adjacent lanes
= [2s + 2π(r + w)] − [2s + 2πr]
= 2πw

So all the organisers need is the lane width w and the value of π. For the standard track, w = 1.22 m, so

stagger = 2 × 3.1416 × 1.22 ≈ 7.67 m
Why it happens: the straight sections cancel because they are common to all lanes, and the two semicircular bends of each lane together make one complete circle. That is why only the difference of the radii, which is the lane width, survives.
Was this helpful? Report an error