Kinematics

In some appropriate units, time (\(t\)) and position (\(x\)) relation of a moving particle is given by \(t = x^2 + x\). The acceleration of the particle is

Two cities X and Y are connected by a regular bus service with a bus leaving in either direction every T min. A girl is driving scooty with a speed of 60 km/h in the direction X to Y notices that a bus goes past her every 30 minutes in the direction of her motion, and every 10 minutes in the opposite direction. Choose the correct option for the period T of the bus service and the speed (assumed constant) of the buses.

A particle moving with uniform speed in a circular path maintains :

The velocity (\(v\)) - time (\(t\)) plot of the motion of a body is shown below :

The acceleration (\(a\)) - time (\(t\)) graph that best suits this motion is :

The angular acceleration of a body, moving along the circumference of a circle, is :

A bullet is fired from a gun at the speed of \(280\,\mathrm{m\,s^{-1}}\) in the direction \(30^\circ\) above the horizontal. The maximum height attained by the bullet is \((g=9.8\,\mathrm{m\,s^{-2}},\ \sin30^\circ=0.5)\) :

A vehicle travels half the distance with speed \(v\) and the remaining distance with speed \(2v\). Its average speed is:

A bullet from a gun is fired on a rectangular wooden block with velocity \(u\). When bullet travels 24 cm through the block along its length horizontally, velocity of bullet becomes \(\frac{u}{3}\). Then it further penetrates into the block in the same direction before coming to rest exactly at the other end of the block. The total length of the block is :

A horizontal bridge is built across a river. A student standing on the bridge throws a small ball vertically upwards with a velocity \(4\,\mathrm{m\,s^{-1}}\). The ball strikes the water surface after 4 s. The height of bridge above water surface is (Take \(g=10\,\mathrm{m\,s^{-2}}\)) :

The ratio of the distances travelled by a freely falling body in the \(1^{st}\), \(2^{nd}\), \(3^{rd}\) and \(4^{th}\) second :

The displacement-time graphs of two moving particles make angles of \(30°\) and \(45°\) with the x-axis as shown in the figure. The ratio of their respective velocity is :

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

A small block slides down on a smooth inclined plane, starting from rest at time t = 0. Let \(S_n\) be the distance travelled by the block in the interval t = n − 1 to t = n. Then, the ratio \(\frac{S_n}{S_{n+1}}\) is :

A car starts from rest and accelerates at 5 m/s\(^2\). 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/s\(^2\))

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 'θ' to the horizontal, the maximum height attained by it equals 4R. The angle of projection, θ, is then given by :

A ball is thrown vertically downward with a velocity of 20 m/s from the top of a tower. It hits the ground after some time with a velocity of 80 m/s. The height of the tower is : (g = 10 m/s\(^2\))

When an object is shot from the bottom of a long smooth inclined plane kept at an angle \(60^\circ\) with horizontal, it can travel a distance \(x_1\) along the plane. But when the inclination is decreased to \(30^\circ\) and the same object is shot with the same velocity, it can travel \(x_2\) distance. Then \(x_1 : x_2\) will be :

The speed of a swimmer in still water is 20 m/s. The speed of river water is 10 m/s and is flowing due east. If he is standing on the south bank and wishes to cross the river along the shortest path, the angle at which he should make his strokes w.r.t. north is given by :

Two particles A and B are moving in uniform circular motion in concentric circles of radii \(r_A\) and \(r_B\) with speed \(v_A\) and \(v_B\) respectively. Their time period of rotation is the same. The ratio of angular speed of A to that of B will be :

Preeti reached the metro station and found that the escalator was not working. She walked up the stationary escalator in time \(t_1\). On other days, if she remains stationary on the moving escalator, then the escalator takes her up in time \(t_2\). The time taken by her to walk up on the moving escalator will be :

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

A particle of mass 10 g moves along a circle of radius 6.4 cm with a constant tangential acceleration. What is the magnitude of this acceleration if the kinetic energy of the particle becomes equal to \(8\times10^{-4}\) J by the end of the second revolution after the beginning of the motion ?

A particle moves so that its position vector is given by \(\vec{r} = \cos\omega t\,\hat{x} + \sin\omega t\,\hat{y}\). 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 :

If the velocity of a particle is \(v = At + Bt^2\), where A and B are constants, then the distance travelled by it between 1s and 2s is :

Two cars \(P\) and \(Q\) start from a point at the same time in a straight line and their positions are represented by \(x_P(t)=at+bt^2\) and \(x_Q(t)=ft-t^2\). At what time do the cars have the same velocity?

In the given figure, \(a=15\) m/s\(^2\) represents the total acceleration of a 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