Question

A mouse is at a distance d from a cat. Both start running with constant equal speed v in such a way that mouse moves in a straight line perpendicular to the initial line joining cat and mouse and cat always aims at the mouse . Separation between them when both move in the same line is

Answer 1

Answer:d/2

Explanation:

Answer 2

Explanation:

Velocity of cat with respect to mouse $=\mathrm{v}-\mathrm{v} \cos \theta$

$\mathrm{v}_{\mathrm{cm}}=\mathrm{v}-\mathrm{v} \cos \theta \rightarrow(1)$

$\mathrm{v}=\frac{\mathrm{dx}}{\mathrm{dt}}$

Since the vertical distance between cat and mouse is decreasing with time,

$\therefore \mathrm{v}_{\mathrm{cm}}=-\frac{\mathrm{d} \mathrm{x}}{\mathrm{dt}}$

$\therefore \mathrm{v}-\mathrm{v} \cos \theta=-\frac{\mathrm{d} \mathrm{x}}{\mathrm{dt}} \rightarrow \mathrm{From}(1)$

$\therefore \int_{0}^{T} v-v \cos \theta . d t=\int_{d}^{0}-d x$

$\therefore \mathrm{VT}-\mathrm{V} \int_{0}^{\mathrm{T}} \cos \theta \cdot \mathrm{dt}=\mathrm{d}$

$\therefore \mathrm{d}=\mathrm{VT}-\mathrm{V} \int_{0}^{\mathrm{T}} \cos \theta \cdot \mathrm{dt}=\mathrm{d} \rightarrow(2)$

Horizontal velocity of cat with respect to mouse $=\mathrm{Vcos} \theta-\mathrm{v}$

$\mathrm{v}_{\mathrm{cm}}=\mathrm{V} \cos \theta-\mathrm{v}$

In time T mouse has covered distance VT

$\therefore \int_{0}^{\mathrm{T}} \mathrm{v}_{\mathrm{cm}} \cdot \mathrm{dt}=\mathrm{VT}$

$\therefore \int_{0}^{T}(V \cos \theta-v) d t=V T$

$\therefore \mathrm{VT}-\mathrm{V} \int_{0}^{\mathrm{T}} \cos \theta \cdot \mathrm{d} \mathrm{t}=\mathrm{VT}$

$\therefore \mathrm{V} \int_{0}^{\mathrm{T}} \cos \theta \cdot \mathrm{dt}=\mathrm{VT}+\mathrm{VT}$

$\therefore V \int_{0}^{T} \cos \theta . d t=2 V T$

$\therefore \int_{0}^{\mathrm{T}} \cos \theta . \mathrm{dt}=2 \mathrm{T} \rightarrow(3)$

From equation (2) and (3)  

$\mathrm{d}=\mathrm{VT}-\mathrm{V}(2 \mathrm{T})$

$\mathrm{d}=\mathrm{VT}-2 \mathrm{VT}$

$\mathrm{d}=-\mathrm{VT}$

$\mathrm{VT}=-\mathrm{d}$

In this time, cat travels d/2  distance. (From the figure)

∴ Distance between cat and mouse at time instant T

$=\frac{\mathrm{d}}{2}-\mathrm{d}$

$=-\frac{\mathrm{d}}{2}$

Distance can never be negative $=\frac{\mathrm{d}}{2}$

$\therefore \text { Separation is } \frac{\mathrm{d}}{2}$