Question

A moving sidewalk in an airport terminal building moves at a speed of $1.0m{s^{ - 1}}$ and is $35.0m$ long. If a woman steps on an one end and walks at $1.5m{s^{ - 1}}$ relative to the moving sidewalk, then find the time that she requires to reach the opposite end:
A) When she walks in the same direction the sidewalk is moving
B) When she walks in an opposite direction

Answer 1

If a woman is moving relative to the moving sidewalk, therefore here the concept of relative velocity is used. Relative velocity is defined as the velocity of an object that is at rest with respect to another frame of reference. In order to describe the complete motion of an object , the concept of relative velocity is used.

Complete step-by-step answer:
Step I:
Given that the velocity of the sidewalk is ${v_{sidewalk}} = 1m{s^{ - 1}}$
Velocity of the women relative to the sidewalk, ${v_{women}} = 1.5m{s^{ - 1}}$
Length of the sidewalk, $d = 35m$
Step II:
When the woman is moving in the direction of the sidewalk, her velocity will increase. The velocity in this case is given by
${V_{total}} = {V_{sidewalk}} + {V_{women}}$
Or ${V_{total}} = 1 + 1.5 = 2.5m{s^{ - 1}}$
Step III:
Velocity is defined as the rate of change of speed with time and its formula is
$V = \dfrac{D}{t}$ ---(i)
Where v is the velocity
D is the displacement or distance covered
T is the time taken
Equation (i) can also be written as
$t = \dfrac{D}{V}$
A) This is the time taken by women to reach the opposite end when she is walking in the direction of the sidewalk.
Distance covered will be equal to the length of the sidewalk, $D = 35m$
Substituting the values,
$t = \dfrac{{35}}{{2.5}} = 14\sec$
Step IV:
B) When the women is moving in the opposite direction of the sidewalk then velocity is given by
${V_{total}} = {V_{women}} - {V_{sidewalk}}$
${V_{total}} = 1.5 - 1 = 0.5m{s^{ - 1}}$
The time taken by women to travel to the opposite end when she is walking in an opposite direction is
$t = \dfrac{{35}}{{0.5}} = 70\sec$

Note: It is to be noted that relative velocity is different from resultant velocity. This is because the relative velocity measures the kinetic energy and the momentum in the frame of reference of the observer. But resultant velocity is the velocity that is obtained by adding the velocities in all dimensions of space in vector form.