The magnitudes of vectors are 3, 4 and 5 units respectively. If , thenthe angle between and is
We have a relation,
Squaring both side, we get
⇒ ( )2 = ()2
⇒( )2+ ()2 + 2 Cosθ = ()2
⇒ (3)2 + (4)2 + 2.3.4 Cosθ = (5)2
⇒ 9 + 16 + 24 Cosθ = 25
⇒ 24 Cosθ = 25
⇒ 24Cosθ = 25 - 25
⇒ θ = Cos-1 (0)
⇒ θ = 90°
The angle between is θ. The value of the triple product is 
Note that . Hence their dot product is zero.
A bus is moving on a straight road towards north with a uniform speed of 50 km/hour turns through 90°. If the speed remains unchanged after turning, the increase in the velocity of bus in the turning process is 
= 50 km h-1 due North;
= 50 km h-1 due West. Angle between
= 50 km h-1 due South
∴ Change in velocity
= 70.7 km/h
The direction of this change in velocity is in South-West.
The maximum range of a gun of horizontal terrain is 16 km. If g = 10 ms–2, then muzzle velocity of a shell must be 
= 16000 [16km = 16000m]
or v = (16000g )½ = (16000 x10)½
= 400 ms –1
Two bodies of same mass are projected with the same velocity at an angle 30° and 60° respectively. The ratio of their horizontal ranges will be
Horizontal range is same when angle of projection is θ or (90° – θ).
An electricfan has blades of length 30cm measured from the axis of rotation. If the fan is rotating at 120 rpm, the acceleration of a point on the tip of the blade is 
When milk is churned, cream gets separated due to
Cream gets separated from a churned milk due to centrifugal force.
The resultant of will be equal to 
When a vector is multiplied with a scalar, the result is a vector.
If a body A of mass M is thrown with velocity v at an angle of 30° to the horizontal and another body B of the same mass is thrown with the same speed at an angle of 60° to the horizontal, the ratio of horizontal range of A to B will be 
Horizontal range is same wh en an gle of projection with the horizonatal is θ and (90° – θ).
A boat is sent across a river with a velocity of 8 km h–1. If the resultant velocity of boat is 10 km h–1 , then the velocity of the river is 
= 6 km h –111.
A body constrained to move in y-direction, is subjected to a force given by . What is the workdone by this force in moving the body through a distance of 10m along y-axis ? 
Since displacement is along the y-direction, hence displacement
When a body moves with a constant speed along a circle
On circular motion, the force acts along the radius and displacement at a location is perpendicular to the radius i.e., θ = 90° As work done =
Th e position vector of a par ticle is . The velocity of the particle is 
Slope of position vector = & slope of velocity vector
∴ velocity is perpendicular to the displacement.
Which of the following is not a vector quantity?
A vector quantity has both magnitude and direction. In the given options, speed has only magnitude, therefore, it is non- vector or scalar quantity.
A bullet is fired from a gun with a speed of 1000 m/s in order to hit a target 100 m away. At what height above the target should the gun be aimed? (The resistance of air is negligible and g = 10 m/s2) 
Speed of the bullet (v) = 1000 m/s and horizontal distance of the target (s) = 100 m.
Time taken to cover the horizontal distance
During this time, the bullet will fall down vertically due to gravitational acceleration.
A body is whirled in a horizontal circle of radius 20 cm. It has an angular velocity of 10 rad/s.What is its linear velocity at any point on circular path
Radius of circular path = 20 cm =
Angular speed of body = 10 rad/s
Linear velocity = radius × Angular speed
A ball whose kinetic energy is E, is thrown at an angle of 45º with the horizontal. Its kinetic energy at the highest point of its flight will be 
Kinetic energy is E
Velocity at highest point
[∴ at the highest point there is only horizontal velocity]
K.E. at highest point
Find the torque of a force acting at the poin . 
Two particles A an d B are con nected by a rigid rod AB. The rod slides along perpendicular rails as shown here. The velocity of A to the left is 10 m/s. What is the velocity of B when angle α = 60º? 
Let after 1 sec angle become 60°. When the end A moves by 10 m left, the end B moves upward by BB' = = 17.3 m/s
A ball of mass 0.25 kg attached to the end of a string of length 1.96 m is moving in a horizontal circle. The string will break if the tension is more than 25 N. What is the maximum speed with which the ball can be moved? 
A 500 kg car takes a round turn of radius 50 m with a velocity of 36 km/h. The centripetal force is
= 1000 N
What is the linear velocity if angular velocity vect or and position vector 
As we know that
The angle between two vectors of magnitude 12 and 18 units when their resultant is 24 units, is
We know that, R2 = A2 + B2 + 2 AB cosθ
(24)2 = (12)2 + (18)2 + 2(12)(18) cos θ
A body of mass 0.4 kg is whi rled in a vertical circle making 2 rev/sec. If the radius of the circle is 1.2 m, then tension in the string when the body is at th e top of the circle, is 
Given : Mass (m) = 0.4 kg Its frequency (n) = 2 rev/sec Radius (r) =1.2 m.
We know that linear velocity of the body (v) = ωr = (2πn)r
= 2 × 3.14 × 1.2 × 2 = 15.08 m/s.
Therefore, tension in the string when the body is at the top of the circle (T)
= 45.78 - 3.92 = 41.56 N
A stone tied with a string, is rotated in a vertical circle. The minimum speed with which the string has to be rotated 
Minimum speed with which the string is rotating in a vertical circle (v) = The minimum speed of stone is independent of mass of stone.
A person swims in a river aiming to reach exactly opposite point on the bank of a river. His speed of swimming is 0.5 m/s at an angle 120º with the direction of flow of water. The speed of water in stream is 
Velocity of person
If a unit vector is represented by , the value of c is 
Two projectiles ar e fir ed from t he same point with the same speed at angles of projection 60º and 30º respectively. Which one of the following is true? 
Given, u1 = u2 = u, θ1 = 60º, θ2 = 30º In 1st case, we know that range
In 2nd case, when θ2 = 30º, then
[we get same value of ranges].
A small sphere is attach ed to a cord and rotates in a vertical circle about a point O. If the average speed of the sphere is increased, the cord is most likely to break at the orientation when the mass is at 
In the case of a body describing a vertical circle,
Tension is maximum when cosθ = +1 and velocity is maximum Both conditions are satisfied at θ = 0º (i.e. at lowest point B]
A particle moves with a velocity under the in fluence of a constant force The instantaneous power applied to the particle is 
= 6 × 20 – 4 × 15 –3 ×5 = 45 J/s