Motion in a Straight Line Set-2

Q1: A rubber ball is released from a height of $5 \ \mathrm{m}$ above the floor. It bounces back repeatedly, always rising to $81 / 100$ of the height through which it falls. Find the average speed of the ball. (Take $g = 10 \ \mathrm{ms}^{-2}$) [(JEE Main 2021)]

Q2: A car accelerates from rest at a constant rate for some time after which it decelerates at a constant rate to come to rest. If the total time elapsed is $t$ seconds, the total distance travelled is: [(JEE Main 2021)]

Q3: A mosquito is moving with a velocity $\vec{v} = 0.5 t^2 \hat{i} + 3 t \hat{j} \ \mathrm{m/s}$ and accelerating in uniform conditions. What will be the direction of mosquito after $2 \ \mathrm{s}$? [(JEE Main 2021)]

Q4: The trajectory of a projectile in a vertical plane is $y = \alpha x - \beta x^2$, where $\alpha$ and $\beta$ are constants and $x \& y$ are respectively the horizontal and vertical distances of the projectile from the point of projection. The angle of projection and the maximum height attained $H$ are respectively given by: [(JEE Main 2021)]

Q5: A scooter accelerates from rest for time $t_1$ at constant rate $a_1$ and then retards at constant rate $a_2$ for time $t_2$ and comes to rest. The correct value of $t_1/t_2$ will be : [(JEE Main 2021)]

Q6: A stone is dropped from the top of a building. When it crosses a point $5 \ \mathrm{m}$ below the top, another stone starts to fall from a point $25 \ \mathrm{m}$ below the top. Both stones reach the bottom of building simultaneously. The height of the building is : [(JEE Main 2021)]

Q7: An engine of a train, moving with uniform acceleration, passes the signal-post with velocity $u$ and the last compartment with velocity $v$. The velocity with which middle point of the train passes the signal post is : [(JEE Main 2021)]

Q8: A particle moves such that it's position vector is $\vec{r} = A (\cos \omega t \hat{i} + \sin \omega t \hat{j})$, where $\omega$ is a constant and $t$ is time. Then which of the following statements is true for the velocity $\vec{v}$ and acceleration $\vec{a}$ of the particle.

Q9: A particle starts from the origin at $t = 0$ with an initial velocity of $\vec{u} = 10 \hat{j} \ \mathrm{cm/s}$ and moves in the $x-y$ plane with a constant acceleration $\vec{a} = (8 \hat{i} + 2 \hat{j}) \ \mathrm{cm/s}^2$. The $x$-coordinate of the particle at the instant when its $y$-coordinate is $32 \ \mathrm{m}$ is $D \ \mathrm{m}$. The value of $D$ is [Note: Units seem inconsistent in source, solving based on $32 \ \mathrm{m}$ in $\mathrm{cm/s}$ context as given.]

Q10: In a car race on straight road, car A takes a time $t$ less than car B at the finish and passes finishing point with a speed $v$ more than that of car B. Both the cars start from rest and travel with constant acceleration $a_1$ and $a_2$, respectively. Then $v$ is equal to [JEE Main 2019]

Q11: A passenger train of length $60 \ \mathrm{m}$ travels at a speed of $80 \ \mathrm{km/hr}$. Another freight train of length $120 \ \mathrm{m}$ travels at a speed of $30 \ \mathrm{km/hr}$. The ratio of times taken by the passenger train to completely cross the freight train when (i) they are moving in same direction and (ii) in the opposite directions is [JEE Main 2019]

Q12: A particle is moving with a velocity $\vec{v} = K (y \hat{i} + x \hat{j})$, where K is a constant. The general equation for its path is (JEE Main 2019)

Q13: The position coordinates of a particle moving in a 3D coordinate system is given by $x = a \cos \omega t$, $y = a \sin \omega t$, and $z = a \omega t$. The speed of the particle is (JEE Main 2019)

Q14: In the cube of side $a$, as shown in the figure, the vector from the central point of the face ABOD to the central point of the face BEFO will be (JEE Main 2019)

Q15: Two guns A and B can fire bullets at speeds $1 \ \mathrm{km/s}$ and $2 \ \mathrm{km/s}$ respectively. From a point on a horizontal ground, they are fired in all possible directions. The ratio of maximum areas covered by the bullets fired by the two guns, on the ground is (JEE Main 2019)

Q16: Two vectors $\vec{A}$ and $\vec{B}$ have equal magnitudes. The magnitude of $(\vec{A} + \vec{B})$ is $n$ times the magnitude of $(\vec{A} - \vec{B})$. The angle between $\vec{A}$ and $\vec{B}$ is (JEE Main 2019)

Q17: A body is projected at $t = 0$ with a velocity $10 \ \mathrm{m/s}$ at an angle of $60^\circ$ with the horizontal. The radius of curvature of its trajectory at $t = 1 \ \mathrm{s}$ is R. Neglecting air resistance and taking acceleration due to gravity $g = 10 \ \mathrm{m/s}^2$, the value of R is (JEE Main 2019)

Q18: A particle moves from the point $(1 \hat{i} + 2 \hat{j}) \ \mathrm{m}$ at $t = 0$ with an initial velocity $\vec{v} = (5 \hat{i} + 4 \hat{j}) \ \mathrm{m/s}$. It is acted upon by a constant force which produces a constant acceleration $\vec{a} = (3 \hat{i} + 3 \hat{j}) \ \mathrm{m/s}^2$. What is the distance of the particle from the origin at time $2 \ \mathrm{s}$? (JEE Main 2019)

Q19: An automobile, travelling at $40 \ \mathrm{km/h}$, can be stopped at a distance of $40 \ \mathrm{m}$ by applying brakes. If the same automobile is travelling at $80 \ \mathrm{km/h}$, the minimum stopping distance, in metres, is (assume no skidding): (JEE Main 2018)

Q20: A car is standing $200 \ \mathrm{m}$ behind a bus, which is also at rest. The two start moving at the same instant but with different forward accelerations. The bus has acceleration $2 \ \mathrm{m/s}^2$ and the car has acceleration $4 \ \mathrm{m/s}^2$. The car will catch up with the bus after a time of (JEE Main 2017)

Q21: A piece of wood of mass $0.03 \ \mathrm{kg}$ is dropped from the top of a $100 \ \mathrm{m}$ height building. At the same time, a bullet of mass $0.02 \ \mathrm{kg}$ is fired vertically upward, with a velocity $100 \ \mathrm{m/s}$, from the ground. The bullet gets embedded in the wood. Then the maximum height to which the combined system reaches above the top of the building before falling below is ($g = 10 \ \mathrm{m/s}^2$)

Q22: A particle has an initial velocity $3 \hat{i} + 4 \hat{j}$ and an acceleration of $0.4 \hat{i} + 0.3 \hat{j}$. Its speed after $10 \ \mathrm{s}$ is

Q23: A parachutist after bailing out falls $50 \ \mathrm{m}$ without friction. When the parachute opens, it decelerates at $2 \ \mathrm{m/s}^2$. He reaches the ground with a speed of $3 \ \mathrm{m/s}$. At what height, did he bailout?

Q24: A car, starting from rest, accelerates at the rate $f$ through a distance $s$, then continues at a constant speed for time $t$ and then decelerates at the rate $f/2$ to come to rest. If the total distance traversed is $S_{total}$, then $s$ is proportional to (Note: Based on the proportional relationship derived in the source, ignoring the given $15 \ \mathrm{s}$ total distance value which is inconsistent with the derived formula).

Q25: An automobile travelling with a speed of $60 \ \mathrm{km/h}$, can brake to stop within a distance of $20 \ \mathrm{m}$. If the car is going twice as fast, i.e., $120 \ \mathrm{km/h}$, the stopping distance will be

Q26: A ball is released from the top of a tower of height $h$ metre. It takes $T$ second to reach the ground. What is the position of the ball in $T/3$ second?

Q27: Which of the following statements is false for a particle moving in a circle with a constant angular speed?

Q28: From a building, two balls A and B are thrown such that A is thrown upwards and B downwards (both vertically). If $v_A$ and $v_B$ are their respective velocities on reaching the ground, then

Q29: If a body loses half of its velocity on penetrating $3 \ \mathrm{cm}$ in a wooden block, then how much will it penetrate more before coming to rest?

Q30: From a tower of height H, a particle is thrown vertically upwards with a speed $u$. The time taken by the particle, to hit the ground is $n$ times that taken by it to reach the highest point of its path. The relation between $H, u$ and $n$ is