Motion in a Plane

Let \(\omega_1, \omega_2\) and \(\omega_3\) be the angular speed of the second hand, minute hand and hour hand of a smoothly running analog clock, respectively. If \(x_1, x_2\) and \(x_3\) are their respective angular distances in 1 minute ...

A bob is whirled in a horizontal circle by means of a string at an initial speed of \(10 \mathrm{~rpm}\). If the tension in the string is quadrupled while keeping the radius constant, the new speed is:

A particle is executing uniform circular motion with velocity \(\vec{v}\) and acceleration \(\vec{a}\). Which of the following is true?

A ball is projected from point A with velocity \(20 \mathrm{~m} \mathrm{~s}^{-1}\) at an angle \(60^{\circ}\) to the horizontal direction. At the highest point \(\mathrm{B}\) of the path (as shown in figure), the velocity $$\mathrm{v} \math...

A horizontal bridge is built across a river. A student standing on the bridge throws a small ball vertically upwards with a velocity \(4 \mathrm{~m} \mathrm{~s}^{-1}\). The ball strikes the water surface after \(4 \mathrm{~s}\). The height ...

A bullet is fired from a gun at the speed of \(280 \mathrm{~ms}^{-1}\) in the direction \(30^{\circ}\) above the horizontal. The maximum height attained by the bullet is
\(\left(g=9.8 \mathrm{~ms}^{-2}, \sin 30^{\circ}=0.5\right)\):-

A cricket ball is thrown by a player at a speed of 20 m/s in a direction 30\(^\circ\) above the horizontal. The maximum height attained by the ball during its motion is
(g = 10 m/s2)

A ball is projected with a velocity, 10 ms\(-\)1, at an angle of 60\(^\circ\) with the vertical direction. Its speed at the highest point of its trajectory will be

A car starts from rest and accelerates at 5m/s2. At t = 4 s, a ball is dropped out of a window by a person sitting in the car. What is the velocity and acceleration of the ball at t = 6 s? (Take g = 10 m/s2)

A particle moving in a circle of radius R with a uniform speed takes a time T to complete one revolution. If this particle were projected with the same speed at an angle '\(\theta\)' to the horizontal, the maximum height attained by it equa...

A particle moving with velocity
\(\overrightarrow V\) is acted by three forces shown by the vector triangle PQR. The velocity of the particle will :

When an object is shot from the bottom of a long smooth inclined plane kept at an angle 60o with horizontal, it can travel a distance x1 along the plane. But when the inclination is decreased to 30o and the same object is shot with the same...

The x and y coordinates of the particle at any time are x = 5t \(-\) 2t2 and y = 10t respectively, where x and y are in metres and t in seconds. The acceleration of the particle at t = 2 s is

In the given figure, a = 15 m s\(-\)2 represents the total acceleration of particle moving in the clockwise direction in a circle of radius R = 2.5 m at a given instant of time. The speed of the particle is

A particle moves so that its position vector is given by \(\overrightarrow r = \cos \omega t\,\widehat x + \sin \,\omega t\,\widehat y,\) where \(\omega\) is a constant.
Which of the following is true?

If the magnitude of sum of two vectors is equal to the magnitude of difference of the two vectors, the angle between these vectors is

A ship A is moving Westwards with a speed of 10 km h\(-\)1 and a ship B 100 km South of A, is moving Northwards with a speed of 10 km h\(-\)1. The time after which the distance between them becomes shortest, is

The positions vector of a particle \(\overrightarrow R\) as a function of time is given by \(\overrightarrow R\) = 4sin(2\(\pi\)t)\(\widehat i\) + 4cos(2\(\pi\)t)\(\widehat j\). Where R is in meters, t is in seconds and \(\widehat i\) a...

If vectors \(\overrightarrow A = \cos \omega t\widehat i + \sin \omega t\widehat j\) and \(\overrightarrow B = \cos {{\omega t} \over 2}\widehat i + \sin {{\omega t} \over 2}\widehat j\) are functions of time, then the value of t at which...

A particle is moving such that is position coordinates (x, y) are (2 m, 3 m) at time t = 0, (6 m, 7 m)
at time t = 2 s and (13 m, 14 m) at time t = 5 s.
Average velocity vector \(\left( {{{\overrightarrow v }_{av}}} \right)\) from t = 0 t...

A projectile is fired from the surface of the earth with a velocity of 5 m s\(-\)1 and angle \(\theta\)
with the horizontal. Another projectile fired from another planet with a velocity of 3 m s\(-\)1 at the same angle follows a traject...

Vectors \(\overrightarrow A ,\overrightarrow B\) and \(\overrightarrow C\) are such that \(\overrightarrow A .\overrightarrow B = 0\) and \(\overrightarrow A .\overrightarrow C = 0\). Then the vector parallel to \(\overrightarrow A\) i...

The velocity of a projectile at the initial point A is \(\left( {2\widehat i + 3\widehat j} \right)\) m/s. It's velocity (in m/s) at point B is

The horizontal range and the maximum height of a projectile are equal. The angle of projection of the projectile is

A particle has initial velocity \(\left( {2\overrightarrow i + 3\overrightarrow j } \right)\) and acceleration \(\left( {0.3\overrightarrow i + 0.2\overrightarrow j } \right)\). The magnitude of velocity after 10 seconds will be

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

A particle moves in a circle of radius 5 cm with constant speed and time period 0.2\(\pi\) s. The acceleration of the particle is

A body is moving with velocity 30 m/s towards east . After 10 seconds its velocity becomes 40 m/s towards north. The average acceleration of the body is

A projectile is fired at an angle of 45o with the horizontal. Elevation angle of the projectile at its highest point as seen from the point of projection, is

A particle has initial velocity \(\left( {3\widehat i + 4\widehat j} \right)\) and has acceleration \(\left( {0.4\widehat j + 0.3\widehat j} \right).\) Its speed after 10 s is

Six vectors, \(\overrightarrow a\) through \(\overrightarrow f\) have the magnitudes and directions indicated in the figure. Which of the following statements is true ?

The speed of a projectile at its maximum height is half of its initial speed. The angle of projection is

A particle moves in x-y plane according to rule x = asin\(\omega\)t and y = acos\(\omega\)t. The particle follows

A particle shows distance - time curve as given in this figure. The maximum instaneous velocity of the particle is around the point

A particle starting from the origin (0, 0) moves in a straight line in the (x, y) planes. Its coordinates at a later time are \(\left( {\sqrt 3 ,3} \right)\). The path of the particle makes with the x-axes an angle of

\(\overrightarrow A\) and \(\overrightarrow B\) are two vectors and \(\theta\) is the angle between them, if $$\left| {\overrightarrow A \times \overrightarrow B } \right| = \sqrt 3 \left( {\overrightarrow A .\overrightarrow B } \right...

The vectors \(\overrightarrow A\) and \(\overrightarrow B\) are such that \(\left| {\overrightarrow A + \overrightarrow B } \right| = \left| {\overrightarrow A - \overrightarrow B } \right|.\) The angle between the two vectors is

For angles of projection of a projectile at angle (45o \(-\) \(\theta\)) and (45o + \(\theta\)), the horizontal range described by the projectile are in the ratio of

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 revoluations in 44 seconds, what is the magnitude and direction of acceleration of the stone ?

If a vector \(2\widehat i + 3\widehat j + 8\widehat k\) is perpendicular to the vector \(4\widehat j - 4\widehat i + \alpha \widehat k,\) then the value of \(\alpha\) is

If the angle between the vectors \(\overrightarrow A\) and \(\overrightarrow B\) is \(\theta\), the value of the product \(\left( {\overrightarrow B \times \overrightarrow A } \right).\overrightarrow A\) 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 other in a t...

If \(\left| {\overrightarrow A \times \overrightarrow B } \right| = \sqrt 3 \overrightarrow A .\overrightarrow B\) then the value of \(\left| {\overrightarrow A + \overrightarrow B } \right|\) is

The vector sum of two forces is perpendicular to their vector differences. In that case, the forces

A particle A is dropped from a height and another particle B is projected in horizontal direction with speed of 5/sec from the same height then correct statement is

Two particles having mass M and m are moving in a circular path having radius R and r. If their time period are same then the ratio of angular velocity will be

If \(\left| {\overrightarrow A + \overrightarrow B } \right| = \left| {\overrightarrow A } \right| + \left| {\overrightarrow B } \right|\) then angle between A and B will be

Two projectiles of same mass and with same velocity are thrown at an angle 60o and 30o with the horizontal, then which will remain same

A man is slipping on a frictionless inclined plane and a bag falls down from the same height. Then the velocity of both is related as

The width of river is 1 km. The velocity of boat is 5 km/hr. The boat covered the width of river in shortest time 15 min. Then the velocity of river stream is