Kinematics · JEE Main previous year questions

A body of mass $10\mathrm{kg}$ is projected at an angle of $45^{\circ }$ with the horizontal. The trajectory of the body is observed to pass through a point $(20,10)$. If $\mathrm{T}$ is the time of flight, then its momentum vector, at time $t=\frac{\mathrm{T}}{\sqrt{2}}$, is ____. $[\text{Take }g=10m/{\mathrm{s}}^2]$

1. A.$100\hat{i}+(100\sqrt{2}-200)\hat{j}$
2. B.$100\sqrt{2}\hat{i}+(100-200\sqrt{2})\hat{j}$
3. C.$100\hat{i}+(100-200\sqrt{2})\hat{j}$
4. D.$100\sqrt{2}\hat{i}+(100\sqrt{2}-200)\hat{j}$

If the initial velocity in horizontal direction of a projectile is unit vector $\hat{i}$ and the equation of trajectory is $y=5x(1-x)$. The $y$ component vector of the initial velocity is ____ $\hat{j}$. (Take $g=10m/{\mathrm{s}}^2$)

A ball is projected from the ground with a speed $15{\text{ ms}}^{-1}$ at an angle $\theta$ with horizontal so that its range and maximum height are equal, then '$\tan \theta$' will be equal to :

1. A.$\frac{1}{4}$
2. B.$\frac{1}{2}$
3. C.2
4. D.4

At $t=0$, truck, starting from rest, moves in the positive x-direction at uniform acceleration of $5\mathrm{m}{\mathrm{s}}^{-2}$. At $t=20\mathrm{s}$, a ball is released from the top of the truck. The ball strikes the ground in $1\mathrm{s}$ after the release. The velocity of the ball, when it strikes the ground, will be : (Given $g=10\mathrm{m}{\mathrm{s}}^{-2}$)

1. A.$100\hat{i}-10\hat{j}$
2. B.$10\hat{i}-100\hat{j}$
3. C.$100\hat{i}$
4. D.$-10\hat{j}$

Two projectiles ${\mathrm{P}}_1$ and ${\mathrm{P}}_2$ thrown with speed in the ratio $\sqrt{3}:\sqrt{2}$, attain the same height during their motion. If ${\mathrm{P}}_2$ is thrown at an angle of $60^{\circ }$ with the horizontal, the angle of projection of ${\mathrm{P}}_1$ with horizontal will be :

1. A.$15^{\circ }$
2. B.$30^{\circ }$
3. C.$45^{\circ }$
4. D.$60^{\circ }$

A car covers AB distance with first one-third at velocity ${\mathrm{v}}_1\mathrm{m}{\mathrm{s}}^{-1}$, second one-third at ${\mathrm{v}}_2\mathrm{m}{\mathrm{s}}^{-1}$ and last one-third at ${\mathrm{v}}_3\mathrm{m}{\mathrm{s}}^{-1}$. If ${\mathrm{v}}_3=3{\mathrm{v}}_1$, ${\mathrm{v}}_2=2{\mathrm{v}}_1$ and ${\mathrm{v}}_1=11\mathrm{m}{\mathrm{s}}^{-1}$ then the average velocity of the car is ____ $\mathrm{m}{\mathrm{s}}^{-1}$.

A projectile is launched at an angle '$\alpha$' with the horizontal with a velocity $20\mathrm{m}{\mathrm{s}}^{-1}$. After $10\mathrm{s}$, its inclination with horizontal is '$\beta$'. The value of $\tan \beta$ will be : ($g=10\mathrm{m}{\mathrm{s}}^{-2}$).

1. A.$\tan \alpha +5\sec \alpha$
2. B.$\tan \alpha -5\sec \alpha$
3. C.$2\tan \alpha -5\sec \alpha$
4. D.$2\tan \alpha +5\sec \alpha$

A girl standing on road holds her umbrella at $45^{\circ }$ with the vertical to keep the rain away. If she starts running without umbrella with a speed of $15\sqrt{2}\mathrm{km}{\mathrm{h}}^{-1}$, the rain drops hit her head vertically. The speed of rain drops with respect to the moving girl is :

1. A.$30\mathrm{km}{\mathrm{h}}^{-1}$
2. B.$\frac{25}{\sqrt{2}}\mathrm{km}{\mathrm{h}}^{-1}$
3. C.$\frac{30}{\sqrt{2}}\mathrm{km}{\mathrm{h}}^{-1}$
4. D.$25\mathrm{km}{\mathrm{h}}^{-1}$

An object is thrown vertically upwards. At its maximum height, which of the following quantity becomes zero?

1. A.Momentum
2. B.Potential Energy
3. C.Acceleration
4. D.Force

For a specific wavelength 670 nm of light coming from a galaxy moving with velocity v, the observed wavelength is 670.7 nm. The value of v is :

1. A.$3\times 10^8{\text{ ms}}^{-1}$
2. B.$3\times 10^{10}{\text{ ms}}^{-1}$
3. C.$3.13\times 10^5{\text{ ms}}^{-1}$
4. D.$4.48\times 10^5{\text{ ms}}^{-1}$

The terminal velocity ($v_{\mathrm{t}}$) of the spherical rain drop depends on the radius ($r$) of the spherical rain drop as :

1. A.$r^{1/2}$
2. B.$r$
3. C.$r^2$
4. D.$r^3$

Given below are two statements. One is labelled as Assertion A and the other is labelled as Reason R. Assertion A : Two identical balls A and B thrown with same velocity 'u' at two different angles with horizontal attained the same range R. If A and B reached the maximum height $h_1$ and $h_2$ respectively, then $R=4\sqrt{h_1{h}_2}$ Reason R : Product of said heights. $h_1{h}_2=\left(\frac{u^2{\sin }^2\theta }{2g}\right)\cdot \left(\frac{u^2{\cos }^2\theta }{2g}\right)$ Choose the correct answer :

1. A.Both A and R are true and R is the correct explanation of A.
2. B.Both A and R are true but R is NOT the correct explanation of A.
3. C.A is true but R is false.
4. D.A is false but R is true.

For a particle in uniform circular motion, the acceleration $\vec{a}$ at any point $\mathrm{P}(R,\theta )$ on the circular path of radius R is (when $\theta$ is measured from the positive x-axis and v is uniform speed) :

1. A.$-\frac{v^2}{R}\sin \theta \hat{i}+\frac{v^2}{R}\cos \theta \hat{j}$
2. B.$-\frac{v^2}{R}\cos \theta \hat{i}+\frac{v^2}{R}\sin \theta \hat{j}$
3. C.$-\frac{v^2}{R}\cos \theta \hat{i}-\frac{v^2}{R}\sin \theta \hat{j}$
4. D.$-\frac{v^2}{R}\hat{i}+\frac{v^2}{R}\hat{j}$

A long cylindrical volume contains a uniformly distributed charge of density $\rho$. The radius of cylindrical volume is R. A charge particle (q) revolves around the cylinder in a circular path. The kinetic energy of the particle is :

1. A.$\frac{\rho q{R}^2}{4{\epsilon }_0}$
2. B.$\frac{\rho q{R}^2}{2{\epsilon }_0}$
3. C.$\frac{q\rho }{4{\epsilon }_0{R}^2}$
4. D.$\frac{4{\epsilon }_0{R}^2}{q\rho }$

A body is projected from the ground at an angle of $45^{\circ }$ with the horizontal. Its velocity after $2\mathrm{s}$ is $20\mathrm{m}{\mathrm{s}}^{-1}$. The maximum height reached by the body during its motion is ____ $\mathrm{m}$. (use $\mathrm{g}=10\mathrm{m}{\mathrm{s}}^{-2}$)

The ranges and heights for two projectiles projected with the same initial velocity at angles $42^{\circ }$ and $48^{\circ }$ with the horizontal are ${\mathrm{R}}_1$, ${\mathrm{R}}_2$ and ${\mathrm{H}}_1$, ${\mathrm{H}}_2$ respectively. Choose the correct option :

1. A.${\mathrm{R}}_1>{\mathrm{R}}_2$ and ${\mathrm{H}}_1={\mathrm{H}}_2$
2. B.${\mathrm{R}}_1={\mathrm{R}}_2$ and ${\mathrm{H}}_1<{\mathrm{H}}_2$
3. C.${\mathrm{R}}_1<{\mathrm{R}}_2$ and ${\mathrm{H}}_1<{\mathrm{H}}_2$
4. D.${\mathrm{R}}_1={\mathrm{R}}_2$ and ${\mathrm{H}}_1={\mathrm{H}}_2$

A particle of mass m is suspended from a ceiling through a string of length L. The particle moves in a horizontal circle of radius r such that $r=\frac{L}{\sqrt{2}}$. The speed of particle will be :

1. A.$\sqrt{rg}$
2. B.$\sqrt{2rg}$
3. C.$2\sqrt{rg}$
4. D.$\sqrt{\frac{rg}{2}}$

A body of mass 2 kg moving with a speed of 4 m/s makes an elastic collision with another body at rest and continues to move in the original direction but with one fourth of its initial speed. The speed of the two body centre of mass is $\frac{x}{10}$ m/s. Then the value of x is ____.

A force $\vec{F}=(40\hat{i}+10\hat{j})N$ acts on a body of mass 5 kg. If the body starts from rest, its position vector $\vec{r}$ at time t = 10 s, will be :

1. A.$(100\hat{i}+400\hat{j})m$
2. B.$(100\hat{i}+100\hat{j})m$
3. C.$(400\hat{i}+100\hat{j})m$
4. D.$(400\hat{i}+400\hat{j})m$

1. A.$C=\sqrt{\frac{2v(0{)}^2}{{\omega }^2}+x(0{)}^2},\varphi ={\tan }^{-1}\left(\frac{x(0)\omega }{2v(0)}\right)$
2. B.$C=\sqrt{\frac{v(0{)}^2}{{\omega }^2}+x(0{)}^2},\varphi ={\tan }^{-1}\left(\frac{x(0)\omega }{v(0)}\right)$
3. C.$C=\sqrt{\frac{v(0{)}^2}{{\omega }^2}+x(0{)}^2},\varphi ={\tan }^{-1}\left(\frac{v(0)}{x(0)\omega }\right)$
4. D.$C=\sqrt{\frac{2v(0{)}^2}{{\omega }^2}+x(0{)}^2},\varphi ={\tan }^{-1}\left(\frac{v(0)}{x(0)\omega }\right)$

A butterfly is flying with a velocity $4\sqrt{2}\text{ m/s}$ in North-East direction. Wind is slowly blowing at $1\text{ m/s}$ from North to South. The resultant displacement of the butterfly in 3 seconds is :

1. A.$12\sqrt{2}\text{ m}$
2. B.$20\text{ m}$
3. C.$3\text{ m}$
4. D.$15\text{ m}$

A person is swimming with a speed of 10 m/s at an angle of $120^{\circ }$ with the flow and reaches to a point directly opposite on the other side of the river. The speed of the flow is 'x' m/s. The value of 'x' to the nearest integer is _______.