A boat which has a speed of 5 km/hr in still water crosses a river of width 1 km along the shortest possible path in 15 minutes. The velocity of the river water in km/hr is [1998, 2000]
Speed along the shortest path
Speed of water
A child is swinging a swing. Min imum and maximum heights of swing from earth’s surface are 0.75 m and 2m respectively. The maximum velocity of this swing is 
Two particles of mass M and m are moving in a circle of radii R and r. If their time-periods are same, what will be the ratio of their linear velocities? 
Linear velocity v = rω
[ω is same in both cases because time period is same]
The angle between the two vectors will be
From a 10 m high building a stone 'A' is dropped, and simultaneously another identical stone 'B' is thrown horizontally with an initial speed of 5 ms–1. Which one of the following statements is true?
h and g are same for both the balls, so time of fall ‘t’ will also be the same for both of them (h is vertical height]
A body of 3 kg moves in the XY plane under the action of a force given by . Assuming that the body is at rest at time t = 0, the velocity of the body at t = 3s is 
The vector sum of two forces is perpendicular to their vector differences. In that case, the forces
A particle moves along a circle of radius with constant tangential acceleration. If the velocity of the particle is 80 m/s at the end of the second revolution after motion has begun, the tangential acceleration is 
Distance travelled in 2 revolutions = 2 × 40 = 80 m
Initial velocity = u = 0
Final velocity v = 80m/sec
Applying the formula, v2 = u2 + 2as
(80)2 = 02 + 2 × a × 80 ⇒ a = 40 m/sec2
If then the value of is
A stone is tied to a string of length ℓ and is whirled in a vertical circle with the other end of the string as the centre. At a certain instant of time, the stone is at its lowest position and has a speed u.
The magnitude of the change in velocity as it reaches a position where the string is horizontal (g being acceleration due to gravity) is 
If a vector is perpendicular to the vector then the value of α is 
For two vectors to be perpendicular to each other
If the angle between the vectors the value of the product is equal to 
Two boys are standing at the ends A and B of a ground where AB = a. The boy at B starts running in a direction perpendicular to AB with velocity v1. The boy at A starts running simultaneously with velocity v and catches the oth er boy in a time t, where t is 
Velocity of A relative to B is given by
By taking x-components of equation (1), we get
By taking Y-components of equation (1), we get
Time taken by boy at A to catch the boy at B is given by
[From equation (1)]
A stone tied to the end of a string of 1 m long is whirled in a horizontal circle with a constant speed. If the stone makes 22 revolutions in 44 seconds, what is the magnitude and direction of acceleration of the stone? 
and direction along the radius towards the centre.
The cir cular motion of a particle with constant speed is 
In circular motion of a particle with constant speed, particle repeats its motion after a regular interval of time but does not oscillate about a fixed point. So, motion of particle is periodic but not simple harmonic.
A car runs at a constant speed on a circular track of radius 100 m, taking 62.8 seconds in every circular loop. The average velocity and average speed for each circular loop respectively, is 
Distance covered in one circular loop = 2πr = 2 × 3.14 × 100 = 628 m
Displacement in one circular loop = 0
For angles of projection of a projectile (45° – θ) and (45° + θ), the horizontal ranges described by the projectile are in the ratio of 
(45º – θ) & (45º + θ) are complementary angles as 45º – θ + 45º + θ = 90º. We know that if angle of projection of two projectiles make complementary angles, their ranges are equal.
In this case also, the range will be same. So the ratio is 1 : 1.
The vectors are such that The angle between the two vectors is [1991, 1996, 2001, 2006]
So, angle between A & B is 90º.
A particle starting from the origin (0, 0) moves in a straight line in the (x, y) plane. Its coordinates at a later time are . The path of the particle makes with the x-axis an angle of 
Let θ be the angle which the particle makes with x axis.
are two vectors and θ is the angle between them, if , the valueof θ is
Three forces acting on a body are shown in the figure. To have the resultant force only along the y- direction, the magnitude of the minimum additional force needed is: 
Th e componen ts of 1N and 2N forces along + x axis = 1 cos60° + 2 sin30°
The component of 4 N force along –x-axis
Therefore, if a force of 0.5N is applied along + x-axis, the resultant force along x-axis will become zero and the resultant force will be obtained only along y-axis.
A particle of mass m is projected with velocity v making an angle of 45° with the horizontal. When the particle lands on the level ground the magnitude of the change in its momentum will be: 
The magnitude of the resultant velocity at the point of projection and the landing point is same.
Clearly, change in momentum along horizontal (i.e along x-axis) = mvcosθ – mv cosθ = 0
Change in momentum along vertical (i.e. along y–axis) = mv sinθ – (–mv sinθ) = 2 mvsinθ = 2mv × sin 45°
Hence, resultant change in momentum
A block of mass m is in contact with the cart C as shown in the Figure. 
The coefficient of static friction between the block and the cart is μ . The acceleration α of the cart that will prevent the block from falling satisfies:
Forces acting on the block are as shown in the fig. Normal reaction N is provided by the force mα due to acceleration α
∴ N = mα For the block not to fall, frictional force,
Six vectors, have the magnitudes and directions indicated in the figure. Which of the following statements is true? 
Using the law of vector addition, is as shown in the fig.
A particle moves in a circle of radius 5 cm with constant speed and time period 0.2πs. The acceleration of the particle is 
Centripetal acceleration ac = ω2r
A missile is fired for maximum range with an initial velocity of 20 m/s. If g = 10 m/s2, the range of the missile is
For maximum range, the angle of projection, θ = 45°.
A projectile is fired at an angle of 45° with the horizontal. Elevation angle of the projectile at its highest point as seen from the point of projection is [2011M]
The horizontal range and the maximum height of a projectile are equal. The angle of projection of the projectiles is : 
According to the problem R = H
A particle ha s initial velocity and acceleration . The magnitude of velocity after 10 seconds will be : 
The velocity of a projectile at the initial point A is m/s. It’s velocity (in m/s) at point B is [NEET 2013]
At point B the direction of velocity component of the projectile along Y - axis reverses.
Vectors are such that and Then the vector parallel to is [NEET Kar. 2013]
Vector triple product