To a person,going eastward in a car with a velocity of $25\, km/hr$,a train appears to move towards north with a velocity of $25\sqrt{3}\, km/hr$. The actual velocity of the train will be..........$km/hr$

An airplane airspeed indicator reads $100 \, m/s$ and its compass shows that it is heading $37^{\circ}$ east of north. The meteorological information provided to the navigator is that the wind velocity is $20 \, m/s$ towards east. The speed of the airplane relative to the ground is closest to ............ $\, m/s$.

Preeti reached the metro station and found that the escalator was not working. She walked up the stationary escalator in time $t_1$. On other days,if she remains stationary on the moving escalator,then the escalator takes her up in time $t_2$. The time taken by her to walk up on the moving escalator will be

Two cars $A$ and $B$ are moving with speeds $v_A = 120 \text{ km/h}$ and $v_B = 50 \text{ km/h}$ respectively in the directions as indicated by the arrows in the figure below. What is the relative speed of car $B$ with respect to car $A$ (in $\text{ km/h}$)?

Two persons $A$ and $B$ are located in the $X-Y$ plane at the points $(0,0)$ and $(0,10)$ respectively. (The distances are measured in $MKS$ units). At time $t=0$, they start moving simultaneously with velocities $\overrightarrow{v}_A = 2\hat{j} \text{ m/s}$ and $\overrightarrow{v}_B = 2\hat{i} \text{ m/s}$ respectively. The time after which $A$ and $B$ are at their closest distance is:

Ship $A$ is moving westwards with a speed of $20 \ km h^{-1}$ and another ship $B$ which is at $200 \ km$ south of $A$ is moving northwards with a speed of $10 \ km h^{-1}$. The time after which the distance between them is shortest and the shortest distance between them are respectively