Kinematics Questions

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 $60\text{km/h}$ in the direction X to Y notices that a bus goes past her every $30\text{minutes}$ in the direction of her motion, and every $10\text{minutes}$ in the opposite direction. Choose the correct option for the period T of the bus service and the speed (assumed constant) of the buses.

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:

The instantaneous angular position of a point on a rotating wheel is given by the equation $\theta (t)=2t^3-6t^2$. The torque on the wheel becomes zero at:

A particle of mass $m$ is released from rest and follows a parabolic path as shown. Assuming that the displacement of the mass from the origin is small, which graph correctly depicts the position of the particle as a function of time?

A missile is fired for maximum range with an initial velocity of $20\text{m/s}$. If $g=10{\text{m/s}}^2$, the range of the missile is:

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

A boy standing at the top of a tower of $20\text{m}$ height drops a stone. Assuming $g=10{\text{ms}}^{-2}$, the velocity with which it hits the ground is:

A stone is dropped from a height $h$. It hits the ground with a certain momentum $P$. If the same stone is dropped from a height $100\%$ more than the previous height, the momentum when it hits the ground will change by:

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

If the velocity of a particle is $v=At+B{t}^2$, where $A$ and $B$ are constants, then the distance travelled by it between $1\text{s}$ and $2\text{s}$ is:

Two particles $A$ and $B$ move with constant velocities ${\vec{v}}_1$ and ${\vec{v}}_2$. At the initial moment their position vectors are ${\vec{r}}_1$ and ${\vec{r}}_2$ respectively. The condition for particles $A$ and $B$ for their collision is:

The position vector of a particle $\vec{R}$ as a function of time is given by:
$$\vec{R}=4\sin (2\pi t)\hat{i}+4\cos (2\pi t)\hat{j}$$
Where $R$ is in meters, $t$ is in seconds and $\hat{i}$ and $\hat{j}$ denote unit vectors along $x$ - and $y$ -directions, respectively. Which one of the following statements is wrong for the motion of particle?

A particle is moving such that its position coordinates $(x,y)$ are $(2\text{m},3\text{m})$ at time $t=0$, $(6\text{m},7\text{m})$ at time $t=2\text{s}$ and $(13\text{m},14\text{m})$ at time $t=5\text{s}$. Average velocity vector $({\vec{V}}_{av})$ from $t=0$ to $t=5\text{s}$ is:

A projectile is fired from the surface of the earth with a velocity of $5\text{m/s}$ and angle $\theta$ with the horizontal. Another projectile fired from another planet with a velocity of $3\text{m/s}$ at the same angle follows a trajectory which is identical with the trajectory of the projectile fired from the earth. The value of the acceleration due to gravity on the planet is (in ${\text{m/s}}^2$): (given $g=9.8{\text{m/s}}^2$)

A stone falls freely under gravity. It covers distances $h_1,h_2$ and $h_3$ in the first $5\text{seconds}$, the next $5\text{seconds}$ and the next $5\text{seconds}$ respectively. The relation between $h_1,h_2$ and $h_3$ is:

The velocity of a projectile at the initial point $A$ is $(2\hat{i}+3\hat{j})\text{m/s}$. Its velocity (in $\text{m/s}$) at point $B$ (the end of the trajectory on the same horizontal plane) is:

The motion of a particle along a straight line is described by equation: $x=8+12t-t^3$ where $x$ is in metre and $t$ in second. The retardation of the particle when its velocity becomes zero, is:

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

A car starts from rest and accelerates so that instantaneous power delivered is constant ($P_0$). The instantaneous velocity is proportional to:

A bus is moving with a speed of $10{\text{ms}}^{-1}$ on a straight road. A scooterist wishes to overtake the bus in $100\text{s}$. If the bus is at a distance of $1\text{km}$ from the scooterist, with what speed should the scooterist chase the bus?

When a ball is thrown up vertically with velocity $v_0$, it reaches a maximum height of $h$. If one wishes to triple the maximum height then the ball should be thrown with velocity:

A body of mass $3\text{kg}$ is under a constant force which causes a displacement $s$ in metres in it, given by the relation $s=\frac{1}{3}{t}^2$, where $t$ is in sec. Work done by the force in $2\text{sec}$ is :

The position $x$ of a particle with respect to time $t$ along the $x$ -axis is given by $x=9t^2-t^3$ where $x$ is in metre and $t$ in second. What will be the position of this particle when it achieves maximum speed along the $+x$ direction?

A particle moving along $x$ -axis has acceleration $f$ at time $t$ given by $f=f_0(1-t/T)$, where $f_0$ and $T$ are constants. The particle at $t=0$ has zero velocity. In the time interval between $t=0$ and the instant when $f=0$, the particle’s velocity ($v_x$) is:

A particle moves in a straight line with a constant acceleration. It changes its velocity from $10{\text{ms}}^{-1}$ to $20{\text{ms}}^{-1}$ while passing through a distance of $135\text{m}$ in $t$ seconds. What is the value of $t$?

A projectile is fired at $45^{\circ }$. Elevation angle of highest point from point of projection is:

A particle covers half of its total distance with speed $v_1$ and the rest half distance with speed $v_2$. Its average speed during the complete journey is:

A ball is dropped from a high rise platform at $t=0$ starting from rest. After $6\text{ seconds}$ another ball is thrown downwards from the same platform with a speed $v$. The two balls meet at $t=18\text{ s}$. What is the value of $v$? (Take $g=10{\text{ m/s}}^2$)

A particle has initial velocity $(3\hat{i}+4\hat{j})$ and has acceleration $(0.4\hat{i}+0.3\hat{j})$. Its speed after $10\text{ s}$ is:

A particle moves a distance $x$ in time $t$ according to equation $x=(t+5{)}^{-1}$. The acceleration of the particle is proportional to:

A particle starts its motion from rest under the action of a constant force. If the distance covered in the first $10$ seconds is $S_1$ and that covered in the first $20$ seconds is $S_2$, then: