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

(1) 25 km/hr

(2) 50 km/hr

(3) 5 km/hr

(4) $5\sqrt{3}$ km/hr

Two particles having position vectors \(\overrightarrow{r_{1}} = \left( 3 \hat{i} + 5 \hat{j}\right)\) metres and \(\overrightarrow{r_{2}} = \left(- 5 \hat{i} - 3 \hat{j} \right)\) metres are moving with velocities \(\overrightarrow{v}_{1} = \left( 4 \hat{i} + 3 \hat{j}\right)\)\(\text{m/s}\) and \(\overrightarrow{v}_{2} = \left(\alpha\hat{i} + 7 \hat{j} \right)\)\(\text{m/s}\). If they collide after \(2\) seconds, the value of \(\alpha\) is:

A batsman hits a sixer and the ball touches the ground outside the cricket field. Which of the following graph describes the variation of the cricket ball's vertical velocity \(v\) with time between the time \(t_1\) as it hits the bat and time \(t_2\) when it touches the ground:

1.		2.
3.		4.

A steam boat goes across a lake and comes back (a) On a quiet day when the water is still and (b) On a rough day when there is a uniform air current so as to help the journey onward and to impede the journey back. If the speed of the launch on both the days was the same, in which case will the steam boat complete the journey in lesser time:

A vector \(\vec{a}\) is turned without a change in its length through a small angle \(d\theta\). The values of \(\left | \Delta \vec{a} \right |\) and \(\Delta a\) are respectively:
1. $\mathrm{}$\(0,\)  \(ad\theta\)
2. \(ad\theta,\) \(0\)
3. \(0,\) \(0\)
4. none of these

A boat crosses a river with a velocity of 8 km/h. If the resulting velocity of boat is 10 km/h, then the velocity of river water is 

(1) 4 km/h

(2) 6 km/h

(3) 8 km/h

(4) 10 km/h

A 120 m long train is moving towards west with a speed of 10 m/s. A bird flying towards east with a speed of 5 m/s crosses the train. The time taken by the bird to cross the train will be 

(1) 16 sec

(2) 12 sec

(3) 10 sec

(4) 8 sec

A boat takes two hours to travel 8 km and back in still water. If the velocity of water is 4 km/h, the time taken for going upstream 8 km and coming back is 

(1) 2h

(2) 2h 40 min

(3) 1h 20 min

(4) Cannot be estimated with the information given

A man sitting in a bus travelling in a direction from west to east with a speed of \(40\) km/h observes that the rain-drops are falling vertically downwards. To another man standing on ground the rain will appear: 

A thief is running away on a straight road in a jeep moving with a speed of 9 m/s. A police man chases him on a motor cycle moving at a speed of 10 m/s. If the instantaneous separation of jeep from the motor cycle is 100 m, how long will it take for the policemen to catch the thief

(1) 1 second

(2) 19 second

(3) 90 second

(4) 100 second