Question

A person walks up a stalled $\mathbf{15}\mathbf{m}$long escalator in$\mathbf{90}\mathit{s}$.When standing on the same escalator, now moving, the person is carried up in$\mathbf{60}\mathbf{s}$. How much time would it take that person to walk up the moving escalator? Does the answer depend on the length of the escalator?

Short answer

(a) The time required to walk up the moving escalator is$36s$

(b) The time does not depend on the length of the escalator.

Step-by-step solution

Given data

• The length of the escalator $\left(\mathrm{l}\right)\mathrm{is}15\mathrm{m}$.
• The time $\left(t_1\right)$man takes to walk the length of the escalator is$90s$.
• The time $\left(t_2\right)$the escalator takes to carry a man is $60s$.

Understanding the Concept

The speed is equal to the distance traveled divided by the time taken. If reference frame A is moving with respect to reference frame B, and reference frame B is also moving with respect to the observer, then the speed of reference A with respect to the observer is equal to the sum of the speed of reference frame A with respect to frame B and speed of reference frame B with respect to the observer.

Using the equation for the speed, we can compare the speed of the escalator and the speed of a man as we know the distance that needs to be covered in the given time. Using the equation for the velocity, and the concept of relative motion, we can find the required quantities.

Formula:

$V=\frac{D}{t}$

Here, V is the speed, and D is the distance traveled in time t.

Find the speeds of the walking person and the elevator

We first determine the speed of the walking person as,

$V_w=\frac{l}{t}$

Substitute the given values.

$\begin{array}{l}\mathrm{V}{}_{\mathrm{w}}=\frac{15\mathrm{m}}{90\mathrm{s}}\\ =0.17\mathrm{m}/\mathrm{s}\end{array}$

Now, as the person stands on the moving escalator, we determine the speed of the escalator as,

$V_e=\frac{l}{t_2}$

Substitute the given values.

$$\begin{gathered} {\mathrm{V}}_{\mathrm{e}}=\frac{15\mathrm{m}}{60\mathrm{s}} \\ =0.25\mathrm{m}/\mathrm{s} \end{gathered}$$

Calculate the velocity of the person with respect to the elevator

The person is now walking on the moving escalator; hence its net speed will be the addition of the two speeds since both the motions are in the same direction.

$V_{we}=V_w+V_e$

Substitute the given values.

$$\begin{gathered} {\mathrm{V}}_{\mathrm{we}}=0.17\mathrm{m}/\mathrm{s} \\ =0.42\mathrm{m}/\mathrm{s} \end{gathered}$$

(a) Calculate the time taken

And then, the time taken for the walk to the escalator is,

$t=\frac{l}{V_{we}}$

Substitute the given values.

$$\begin{gathered} t=\frac{15m}{0.42m/s} \\ =36s \end{gathered}$$

Therefore, the time required to walk up the moving escalator is $36s$.

(b) Finding the dependency on the length of the escalator

The time required to walk up the moving escalator depends on the speed of the walking person and the escalator itself. It does not depend on the length of the escalator.