Question

A moving beltway at an airport has a speed ${\mathit{v}}_{\mathbf{1}}$and a length L. A woman stands on the beltway as it moves from one end to the other, while a man in a hurry to reach his flight walks on the beltway with a speed of${\mathit{y}}_{\mathbf{2}}$relative to the moving beltway.

(a) What time interval is required for the woman to travel the distance L?

(b) What time interval is required for the man to travel this distance?

(c) A second beltway is located next to the first one. It is identical to the first one but moves in the opposite direction at speed${\mathit{v}}_{\mathbf{1}}$. Just as the man steps onto the beginning of the beltway and begins to walk at speed${\mathit{v}}_{\mathbf{2}}$relative to his beltway, a child steps on the other end of the adjacent beltway. The child stands at rest relative to this second beltway. How long after stepping on the beltway does the man pass the child?

Short answer

(a) The time interval required by the women to travel the distance L is$\frac{L}{v_1}$ .

(b) The time interval required by the man to travel the distance L is $\frac{L}{v_1+v_2}$.

(c) The time interval required by the man to pass the child is$\frac{L}{\left(v_1+2v_2\right)}$ .

Step-by-step solution

Identification of the given data

The given data can be listed below as,

• The speed of the beltway is$v_1$
• The length of the beltway is L
• The speed of the men relative to the moving beltway is $v_2$.

From the given situation the women stand on the beltway it means her speed is also$v_1$ .

The formula for time interval

The time interval is given by,

$\mathit{t}\mathbf{=}\frac{\mathbf{D}}{\mathbf{v}}$

Here,

D = distance

v = speed

Determination of the time interval is required for the woman to travel the distance L

(a)

The formula to calculate the time interval required for the distance is,

$t=\frac{L}{v_1}$

Here, t is the time interval required for this distance.

Therefore, the time interval required by women to travel the distance L is$\frac{L}{v_1}$ .

The time interval required by the man to travel the distance

(b)

From the given situation the women stands on the beltway it means her speed is also $v_1$.

The walks on the moving beltway, so the resultant speed of the man relative to the ground is,$v=v_1+v_2$

The formula to calculate the time interval required for the distance is,

$t^{\text{'}}=\frac{L}{v}$

Here, $t^{\text{'}}$ is the time interval required by the man.

Substitute values in the above equation.

$t=\frac{L}{v_1+v_2}$

Therefore, the time interval required by the man to travel the distance L is $\frac{L}{v_1+v_2}$.

The time interval required by the man to pass the child

(c)

From the given situation the child stands at rest on the another belt way and so the speed of the child is $v_2$.

The direction of another beltway is opposite to the first one and so the relative speed of the man while he passes the child is,

$\begin{array}{rcl}{v}_r& =& \left(v_1+v_2\right)-v_1\\ & =& v_2\end{array}$

Consider the distance travelled by man at the time he passes the child is l.

The time interval required by the man for this distance is,

$t=\frac{l}{v_1+v_2}$ ...(1)

The distance covered by the child in the same time is,

$l^{\text{'}}=L-l$

Here, $l^{\text{'}}$is the distance covered by the child.

The time interval required by the child for this distance is, $t=\frac{l^{\text{'}}}{v_2}$

Substitute values in the above equation.

$t=\frac{L-l}{v_2}$ ...(2)

Equate both equations

$\begin{array}{rcl}\frac{l}{v_1+v_2}& =& \frac{L-l}{v_2}\\ l\left(v_2\right)& =& (L-l)\left(v_1+v_2\right)\\ l\left(v_2\right)& =& L\left(v_1+v_2\right)-l\left(v_1+v_2\right)\\ l\left(v_2\right)+l\left(v_1+v_2\right)& =& L\left(v_1+v_2\right)\\ & & \end{array}$

Further solve the above equation.

$\begin{array}{rcl}l\left(v_1+2v_2\right)& =& L\left(v_1+v_2\right)\\ l& =& \frac{L\left(v_1+v_2\right)}{\left(v_1+2v_2\right)}\end{array}$

Thus, the distance travelled by man when he passes child is $\frac{L\left(v_1+v_2\right)}{\left(v_1+2v_2\right)}$.

Substitute values in equation (1).

$$\begin{gathered} t=\frac{\frac{L\left(v_1+v_2\right)}{\left(v_1+2v_2\right)}}{v_1+v_2} \\ =\frac{L}{\left(v_1+2v_2\right)} \end{gathered}$$

Therefore, the time interval required by the man to pass the child is $\frac{L}{\left(v_1+2v_2\right)}$.