JEE Advanced Physics Relative Velocity in One Dimension PYQ

Q 1 :

Airplanes A and B are flying with constant velocity in the same vertical plane at angles $30 °$ and $60 °$ with respect to the horizontal respectively as shown in figure. The speed of A is $100 \sqrt{3} m/s$ . At time $t = 0 s$ , an observer in A finds B at a distance of $500 m$ . The observer sees B moving with a constant velocity perpendicular to the line of motion of A. If at $t = t_{0}$ , A just escapes being hit by B, $t_{0}$ in seconds is [2014]

[IMAGE 15]

(5)

[IMAGE 16]

$As 'A' see the motion of 'B' ⊥ r to {\vec{V}}_{A} .$

$So, v_{A} = v_{B} \cos 30 ° ...(i)$

$Now, {\vec{V}}_{B A} = {\vec{V}}_{B} - {\vec{V}}_{A}$

$= v_{B} \cos 30 ° \hat{i} + v_{B} \sin 30 ° \hat{j} - v_{A} \hat{i}$

$= v_{B} \sin 30 ° \hat{j} [From (i)]$

$= \frac{v_{A}}{\cos 30 °} \sin 30 ° \hat{j} [From (i)]$

$= v_{A} \tan 30 ° \hat{j} = \frac{v_{A}}{\sqrt{3}} = 100 m/s$

$So, t_{0} = \frac{500}{| {\vec{V}}_{B A} |} = \frac{500}{100} = 5 sec .$

Q 2 :

STATEMENT-1: For an observer looking out through the window of a fast moving train, the nearby objects appear to move in the opposite direction to the train, while the distant objects appear to be stationary.

STATEMENT-2: If the observer and the object are moving at velocities ${\vec{v}}_{1}$ and ${\vec{v}}_{2}$ respectively with reference to a laboratory frame, the velocity of the object with respect to the observer is ${\vec{v}}_{2} - {\vec{v}}_{1}$ . [2008]

Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
Statement-1 is True, Statement-2 is False
Statement-1 is False, Statement-2 is True

(2)

For a moving observer, the nearby objects appear to move in the opposite direction at a large speed. This is because the angular speed of the nearby object w.r.t. the observer is large. As the object moves away, the angular velocity decreases and therefore its speed seems to be less. The distant object almost remains stationary.

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$∵$ $θ_{2} > θ_{1}$ $∴$ $ω_{2} > ω_{1}$

$From the concept of relative velocity$

${\vec{V}}_{21} = {\vec{V}}_{2 G} - {\vec{V}}_{1 G}$

$where G is the laboratory frame.$

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