Kinematics — JEE Main PYQs

The two projectiles are projected with the same initial velocities at the $15°$ and $30°$ with respect to the horizontal. The ratio of their ranges is $1{:}x$. The value of $x$ is

Two cars $A$ and $B$ are moving in the same direction along a straight line with speeds $100 \text{ km/h}$ and $80 \text{ km/h}$, respectively such that car $A$ is moving ahead of car $B$. A person in car $B$ throws a stone with a speed $v$ so that it hits the car $A$ with a speed of $5 \text{ m/s}$. The value of $v$ is _______ km/h.

If $x$ and $y$ coordinates of a projectile as a function of time $(t)$ are given as $24t$ and $43.6t - 4.9t^2$, respectively, then the angle (in degrees) made by the projectile with horizontal when $t = 2$ s is _________.

A gun mounted on the ground fires bullets in all directions with same speed. The farthest distance the bullets could reach is $6.4 \text{ m}$. The speed of the bullets from the gun is ______ m/s.

(take $g = 10 \text{ m/s}^2$)

From 18 m height above the ground a ball is dropped from rest. The height above the ground at which the magnitude of velocity equal to the magnitude of acceleration (in the same set of units) due to gravity is ____ m.

(Take $g = 10 \text{ m/s}^2$ and neglect the air resistance)

The velocity ($v$) versus time ($t$) plot of a particle is shown in the figure, for a time interval of $40 \text{ s}$. The total distance travelled by the particle and the average velocity during this period are, respectively ________.

[Figure: velocity-time graph — $v(t)$ in m/s rises linearly from 0 to +5, returns to 0 at $t = 20$ s, dips linearly to −5 and returns to 0 at $t = 40$ s (triangular waveform between +5 and −5 m/s)]

A gas balloon is going up with a constant velocity of $10\,\text{m/s}$. When this balloon reached a height of $75\,\text{m}$, a stone is dropped from it and balloon keeps moving up with the same velocity. The height of the balloon when the stone hits the ground is ________ m. (Take $g = 10\,\text{m/s}^2$)

Two identical bodies, projected with the same speed at two different angles cover the same horizontal range $R$. If the time of flight of these bodies are $5\,\text{s}$ and $10\,\text{s}$, respectively, then the value of $R$ is ________ m. (Take $g = 10\,\text{m/s}^2$)

A river is flowing from west to east direction with speed of $9~\mathrm{km~h^{-1}}$. If a boat capable of moving at a maximum speed of $27~\mathrm{km~h^{-1}}$ in still water, crosses the river in half a minute, while moving with maximum speed at an angle of $150°$ to direction of river flow, then the width of the river is :

A person travelling on a straight line moves with a uniform velocity $v_1$ for a distance $x$ and with a uniform velocity $v_2$ for the next $\dfrac{3}{2}x$ distance. The average velocity in this motion is $\dfrac{50}{7}~\mathrm{m/s}$. If $v_1$ is 5 m/s then $v_2=$_________ m/s.

A sportsman runs around a circular track of radius r such that he traverses the path ABAB. The distance travelled and displacement, respectively, are

[Figure: a circle of radius r with points A and B at the two ends of a horizontal diameter]

The angle of projection of a particle is measured from the vertical axis as $\phi$ and the maximum height reached by the particle is $h_m$. Here $h_m$ as function of $\phi$ can be presented as

Which of the following curves possibly represent one-dimensional motion of a particle?

A. [Figure: phase vs time — a straight line through the origin with positive slope]

B. [Figure: velocity vs displacement — a circle centred at the origin]

C. [Figure: velocity vs time — a circle centred at the origin]

D. [Figure: total distance vs time — a non-decreasing curve rising in the first quadrant]

Choose the correct answer from the options given below:

A particle is projected with velocity $u$ so that its horizontal range is three times the maximum height attained by it. The horizontal range of the projectile is given as $\frac{nu^2}{25g}$, where value of $n$ is: (Given, '$g$' is the acceleration due to gravity.)

A particle moves along the $x$-axis and has its displacement $x$ varying with time $t$ according to the equation:

$$x = c_0(t^2 - 2) + c(t - 2)^2$$

where $c_0$ and $c$ are constants of appropriate dimensions. Then, which of the following statements is correct?

The displacement $x$ versus time graph is shown below.

[Figure: displacement (m) vs time (s) graph; the curve starts at O, dips to A (−5 m) at t≈1 s, returns to zero at B (t=2 s), rises to C (5 m) at t=3 s, stays flat to D (5 m) at t=5 s, rises to peak E (10 m) at t=6 s, falls through F (t=7 s) to G (−5 m) at t≈8 s, then returns to zero at H (t=9 s)]

(A) The average velocity during 0 to 3 s is 10 m/s (B) The average velocity during 3 to 5 s is 0 m/s (C) The instantaneous velocity at t = 2 s is 5 m/s (D) The average velocity during 5 to 7 s and instantaneous velocity at t = 6.5 s are equal (E) The average velocity from t = 0 to t = 9 s is zero

Choose the correct answer from the options given below :

Two projectiles are fired from ground with same initial speeds from same point at angles $(45° + \alpha)$ and $(45° - \alpha)$ with horizontal direction. The ratio of their times of flights is

A helicopter flying horizontally with a speed of 360 km/h at an altitude of 2 km, drops an object at an instant. The object hits the ground at a point O, 20 s after it is dropped. Displacement of 'O' from the position of helicopter where the object was released is :

(use acceleration due to gravity $g = 10\ \mathrm{m/s^2}$ and neglect air resistance)

Two balls with same mass and initial velocity, are projected at different angles in such a way that maximum height reached by first ball is 8 times higher than that of the second ball. $\mathrm{T}_1$ and $\mathrm{T}_2$ are the total flying times of first and second ball, respectively, then the ratio of $\mathrm{T}_1$ and $\mathrm{T}_2$ is

A particle is projected at an angle of 30° from horizontal at a speed of 60 m/s. The height traversed by the particle in the first second is $h_0$ and height traversed in the last second, before it reaches the maximum height, is $h$. The ratio $h_0/h$ is ________. [Note: the final line of this stem is corrupted in the source PDF after "maximum h"; the closing wording is reconstructed]

Two particles are located at equal distance from origin. The position vectors of those are represented by $\vec{A} = 2\hat{i} + 3n\hat{j} + 2\hat{k}$ and $\vec{B} = 2\hat{i} - 2\hat{j} + 4p\hat{k}$, respectively. If both the vectors are at right angle to each other, the value of $n^{-1}$ is _____.

The position vector of a moving body at any instant of time is given as $\vec{r} = \left(5t^2\,\hat{i} - 5t\,\hat{j}\right)\,\mathrm{m}$. The magnitude and direction of velocity at $t = 2\,\mathrm{s}$ is,

The velocity-time graph of an object moving along a straight line is shown in figure. What is the distance covered by the object from $t = 0$ to $t = 4\,\mathrm{s}$?

[Figure: velocity-time graph; $v$ (in $\mathrm{ms^{-1}}$) rises linearly from O at $t=0$ to $10$ at point A ($t=2\,\mathrm{s}$), stays constant at $10$ from A to B ($t=4\,\mathrm{s}$), then decreases linearly to zero at $t=6\,\mathrm{s}$]

The co-ordinates of a particle moving in $x$-$y$ plane are given by : $x = 2 + 4t$, $y = 3t + 8t^2$. The motion of the particle is :

A body travels 102.5 m in $n^{\text{th}}$ second and 115.0 m in $(n+2)^{\text{th}}$ second. The acceleration is :

A cyclist starts from the point $P$ of a circular ground of radius 2 km and travels along its circumference to the point S. The displacement of a cyclist is:

[Figure: a circle with centre O; point P at the top, Q at the right, R at the bottom, S at the left; arrows along the circumference show travel from P through Q and R to S]

A bus moving along a straight highway with speed of $72~km/h$ is brought to halt within 4 $s$ after applying the brakes. The distance travelled by the bus during this time (Assume the retardation is uniform) is ______ $m$.

The angle between vector $\vec{Q}$ and the resultant of $\left(2\vec{Q} + 2\vec{P}\right)$ and $\left(2\vec{Q} - 2\vec{P}\right)$ is :

A body moves on a frictionless plane starting from rest. If $S_n$ is distance moved between $t = n-1$ and $t = n$ and $S_{n-1}$ is distance moved between $t = n-2$ and $t = n-1$, then the ratio $\frac{S_{n-1}}{S_n}$ is $\left(1 - \frac{2}{x}\right)$ for $n = 10$. The value of $x$ is ________.

A man carrying a monkey on his shoulder does cycling smoothly on a circular track of radius $9\ \mathrm{m}$ and completes 120 resolutions in 3 minutes. The magnitude of centripetal acceleration of monkey is (in $\mathrm{m/s^2}$) :

The maximum height reached by a projectile is 64 m. If the initial velocity is halved, the new maximum height of the projectile is ________ m.

A train starting from rest first accelerates uniformly up to a speed of 80 $km/h$ for time $t$, then it moves with a constant speed for time $3t$. The average speed of the train for this duration of journey will be (in $km/h$) :

For three vectors $\vec{A} = (-x\hat{i} - 6\hat{j} - 2\hat{k})$, $\vec{B} = (-\hat{i} + 4\hat{j} + 3\hat{k})$ and $\vec{C} = (-8\hat{i} - \hat{j} + 3\hat{k})$, if $\vec{A}\cdot(\vec{B}\times\vec{C}) = 0$, then value of $x$ is _________.

A body projected vertically upwards with a certain speed from the top of a tower reaches the ground in $t_1$. If it is projected vertically downwards from the same point with the same speed, it reaches the ground in $t_2$. Time required to reach the ground, if it is dropped from the top of the tower, is :

A particle moves in a straight line so that its displacement $x$ at any time $t$ is given by $x^2 = 1 + t^2$. Its acceleration at any time $t$ is $x^{-n}$ where $n =$ ________.

A clock has 75 cm, 60 cm long second hand and minute hand respectively. In 30 minutes duration the tip of second hand will travel $x$ distance more than the tip of minute hand. The value of $x$ in meter is nearly (Take $\pi = 3.14$) :

Three vectors $\vec{OP}, \vec{OQ}$ and $\vec{OR}$ each of magnitude A are acting as shown in figure. The resultant of the three vectors is $A\sqrt{x}$. The value of $x$ is ________.

[Figure: circle centred at O with three radial vectors — OP along the positive x-axis, OQ at 90° from OP (vertically up), and OR at 45° below OP]

The angle of projection for a projectile to have same horizontal range and maximum height is :

A body of mass M thrown horizontally with velocity $v$ from the top of the tower of height H touches the ground at a distance of 100 m from the foot of the tower. A body of mass 2M thrown at a velocity $\frac{v}{2}$ from the top of the tower of height 4H will touch the ground at a distance of ________ m.

A particle moving in a straight line covers half the distance with speed 6 m/s. The other half is covered in two equal time intervals with speeds 9 m/s and 15 m/s respectively. The average speed of the particle during the motion is :

If $\vec{a}$ and $\vec{b}$ makes an angle $\cos^{-1}\left(\dfrac{5}{9}\right)$ with each other, then $\left|\vec{a} + \vec{b}\right| = \sqrt{2}\left|\vec{a} - \vec{b}\right|$ for $\left|\vec{a}\right| = n\left|\vec{b}\right|$. The integer value of n is ________.

Two cars are travelling towards each other at speed of $20~\mathrm{m~s^{-1}}$ each. When the cars are 300 m apart, both the drivers apply brakes and the cars retard at the rate of $2~\mathrm{m~s^{-2}}$. The distance between them when they come to rest is :

The resultant of two vectors $\vec{A}$ and $\vec{B}$ is perpendicular to $\vec{A}$ and its magnitude is half that of $\vec{B}$. The angle between vectors $\vec{A}$ and $\vec{B}$ is ________ $°$.

A particle moving in a circle of radius R with uniform speed takes time T to complete one revolution. If this particle is projected with the same speed at an angle $\theta$ to the horizontal, the maximum height attained by it is equal to 4R. The angle of projection $\theta$ is then given by :

A particle is moving in one dimension (along $x$ axis) under the action of a variable force. It's initial position was 16 m right of origin. The variation of its position ($x$) with time (t) is given as $x = -3t^3 + 18t^2 + 16t$, where $x$ is in m and t is in s. The velocity of the particle when its acceleration becomes zero is ________ m/s.

Train A is moving along two parallel rail tracks towards north with speed 72 km/h and train B is moving towards south with speed 108 km/h. Velocity of train B with respect to A and velocity of ground with respect to B are (in $\mathrm{ms}^{-1}$) :

A particle initially at rest starts moving from reference point $x = 0$ along $x$-axis, with velocity $v$ that varies as $v = 4\sqrt{x}$ m/s. The acceleration of the particle is ________ $\mathrm{ms}^{-2}$.

Position of an ant (S in metres) moving in Y-Z plane is given by $S = 2t^2\hat{j} + 5\hat{k}$ (where t is in second). The magnitude and direction of velocity of the ant at $t = 1$ s will be :

A particle starts from origin at $t = 0$ with a velocity $5\hat{i}$ m/s and moves in $x$-$y$ plane under action of a force which produces a constant acceleration of $(3\hat{i} + 2\hat{j})\ \mathrm{m/s^2}$. If the $x$-coordinate of the particle at that instant is 84 m, then the speed of the particle at this time is $\sqrt{\alpha}$ m/s. The value of $\alpha$ is ________.

A body starts moving from rest with constant acceleration covers displacement $S_1$ in first $(p - 1)$ seconds and $S_2$ in first $p$ seconds. The displacement $S_1 + S_2$ will be made in time :

A ball rolls off the top of a stairway with horizontal velocity $u$. The steps are 0.1 m high and 0.1 m wide. The minimum velocity $u$ with which that ball just hits the step 5 of the stairway will be $\sqrt{x}\ \mathrm{ms}^{-1}$ where $x =$ __________ [use $\mathrm{g} = 10\ \mathrm{m/s^2}$].

The displacement and the increase in the velocity of a moving particle in the time interval of t to $(t+1)$ s are 125 m and 50 m/s, respectively. The distance travelled by the particle in $(t+2)^{\text{th}}$ s is ________ m.

Projectiles A and B are thrown at angles of $45°$ and $60°$ with vertical respectively from top of a $400$ m high tower. If their ranges and times of flight are same, the ratio of their speeds of projection $v_{\mathrm{A}} : v_{\mathrm{B}}$ is :

[Take $g = 10\ ms^{-2}$]

A vector has magnitude same as that of $\vec{A} = 3\hat{i} + 4\hat{j}$ and is parallel to $\vec{B} = 4\hat{i} + 3\hat{j}$. The $x$ and $y$ components of this vector in first quadrant are $x$ and $3$ respectively where $x =$ ________.

The relation between time $'t'$ and distance $'x'$ is $t = \alpha x^2 + \beta x$, where $\alpha$ and $\beta$ are constants. The relation between acceleration $(a)$ and velocity $(v)$ is :

A body starts falling freely from height $H$ hits an inclined plane in its path at height $h$. As a result of this perfectly elastic impact, the direction of the velocity of the body becomes horizontal. The value of $\frac{H}{h}$ for which the body will take the maximum time to reach the ground is ________.

If two vectors $\vec{A}$ and $\vec{B}$ having equal magnitude $R$ are inclined at an angle $\theta$, then