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
4 km/hr
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 [2001]
Two particles of mass M and m are moving in a circle of radii R and r. If their timeperiods are same, what will be the ratio of their linear velocities? [2001]
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?[2002]
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 [2002]
The vector sum of two forces is perpendicular to their vector differences. In that case, the forces[2003]
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 [2003]
Circumference
Distance travelled in 2 revolutions = 2 × 40 = 80 m
Initial velocity = u = 0
Final velocity v = 80m/sec
Applying the formula, v^{2} = u^{2} + 2as
(80)^{2} = 02 + 2 × a × 80 ⇒ a = 40 m/sec^{2}
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 [2004]
If a vector is perpendicular to the vector then the value of α is [2005]
For two vectors to be perpendicular to each other
If the angle between the vectors the value of the product is equal to [2005]
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 v_{1}. The boy at A starts running simultaneously with velocity v and catches the oth er boy in a time t, where t is [2005]
Velocity of A relative to B is given by
By taking xcomponents of equation (1), we get
By taking Ycomponents of equation (1), we get
.....(3)
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? [2005]
and direction along the radius towards the centre.
The cir cular motion of a particle with constant speed is [2005]
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 [2006]
Distance covered in one circular loop = 2πr = 2 × 3.14 × 100 = 628 m
Speed
Displacement in one circular loop = 0
Velocity
For angles of projection of a projectile (45° – θ) and (45° + θ), the horizontal ranges described by the projectile are in the ratio of [2006]
(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 xaxis an angle of [2007]
Let θ be the angle which the particle makes with x axis.
From figure,
are two vectors and θ is the angle between them, if , the valueof θ is[2007]
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: [2008]
Th e componen ts of 1N and 2N forces along + x axis = 1 cos60° + 2 sin30°
The component of 4 N force along –xaxis
Therefore, if a force of 0.5N is applied along + xaxis, the resultant force along xaxis will become zero and the resultant force will be obtained only along yaxis.
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: [2008]
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 xaxis) = 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. [2010]
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? [2010]
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 [2011]
Centripetal acceleration a_{c} = ω^{2}r
A missile is fired for maximum range with an initial velocity of 20 m/s. If g = 10 m/s^{2}, 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 : [2012]
Horizontal range
....(1)
Maximum height
....(2)
According to the problem R = H
A particle ha s initial velocity and acceleration . The magnitude of velocity after 10 seconds will be : [2012]
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
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