Question

A bus is moving with a speed of $10m{s^{ - 1}}$ on a straight road. A scooterist wishes to overtake the bus in 100s. If the bus is at a distance of 1Km from the scooterist with what speed should the scooterist chase the bus?
$
  (a){\text{ 40m}}{{\text{s}}^{ - 1}} \\
  (b){\text{ 25m}}{{\text{s}}^{ - 1}} \\
  (c){\text{ 10m}}{{\text{s}}^{ - 1}} \\
  (d){\text{ 20m}}{{\text{s}}^{ - 1}} \\
$

Answer 1

Hint – In this question let the speed of the scooter be ${V_s}$ m/s. Then use the concept of relative velocity, that is relative speed of the scooterist and the bus is $\left( {{V_s} - {V_b}} \right)$ m/s, where ${V_b}$ is the speed of the bus. Then use the relation between the distances, time and speed that is $dist = speed \times time$. This will help to approach the problem.
Step by step answer:

Speed (${V_b}$) of the bus = 10m/s
Distance between the scooterist and the bus = 1Km
As we know 1Km = 1000m.
Therefore, distance (d) between the scooterist and the bus = 1000m.
Now the scooterist wishes to overtake the bus in 100 seconds.
So the speed of the scooterist must be greater than the speed of the bus.
Let the speed of the scooterist = ${V_s}$ m/s.
Now according to the theory of relativity that if two bodies travel with different speeds in the same direction then the relative speed is the difference of the individual speeds.
And if the two bodies travel with different speeds in opposite directions than the relative speed is the sum of the individual speeds.
So the scooter overtakes the bus therefore both the scooter and the bus travel in the same direction.
So the relative speed of the scooterist and the bus is $\left( {{V_s} - {V_b}} \right)$ m/s.
And the relative time in which the scooter overtakes the bus = t = 100 s.
Now as we know the relation between speed, distance and time, speed is the ratio of distance travelled to the time taken.
$ \Rightarrow {V_s} - {V_b} = \dfrac{d}{t}$
 Now substitute the values we have,
$ \Rightarrow {V_s} - 10 = \dfrac{{1000}}{{100}}$
Now simplify this we have,
$ \Rightarrow {V_s} = 10 + 10 = 20$ m/s.
So the speed of the scooterist is 20 m/s in which the scooter overtakes the bus.
So this is the required answer.
Hence option (D) is the correct answer.

Note – The concept of relativity is very helpful in solving problems of this kind. We must have come face to face with problems like two trains travelling opposite to each other at certain velocities, at what time they cross each other. In them too the concept of relativity comes into play, we simply make one object stationary by giving its speed in exactly the opposite direction to the other object. This forms the basis of relativity.