Motion in a Plane Set-2

Q1: The position vector of a particle changes with time according to the relation $\vec{r}(t) = 15t^2 \hat{i} + (4 - 20t^2)\hat{j}$. What is the magnitude of the acceleration at $t = 1$?

Q2: A particle moves from the point $(2.0\hat{i} + 4.0\hat{j}) m$, at $t = 0$, with an initial velocity $(5.0\hat{i} + 4.0\hat{j}) \text{ms}^{-1}$. It is acted upon by a constant force which produces a constant acceleration $(4.0\hat{i} + 4.0\hat{j}) \text{ms}^{-2}$. What is the distance of the particle from the origin at time $2s$?

Q3: A particle is moving along the $x$-axis with its coordinate with time ‘$t$’ given by $x(t) = 10 + 8t - 3t^2$. Another particle is moving along the $y$-axis with its coordinate as function of time given by $y(t) = 5 - 8t^3$. At $t = 1 s$, the speed of the second particle as measured in the frame of the first particle is given as $\sqrt{v}$. Then $v$ (in $\text{m/s}$) is ______.

Q4: The trajectory of a projectile near the surface of the earth is given as $y = 2x - 9x^2$. If it were launched at an angle $\theta_0$ with speed $v_0$ then ($g = 10 \text{ms}^{-2}$)

Q5: Two guns A and B can fire bullets at speed $1 \text{km/s}$ and $2 \text{km/s}$ 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:

Q6: The position of a projectile launched from the origin at $t = 0$ is given by $\vec{r} = (40\hat{i} + 50\hat{j}) m$ at $t = 2s$. If the projectile was launched at an angle $\theta$ from the horizontal, then $\theta$ is (take $g = 10 \text{ms}^{-2}$)

Q7: A projectile is given an initial velocity of $(\hat{i} + 2\hat{j}) \text{m/s}$, where $\hat{i}$ is along the ground and $\hat{j}$ is along the vertical. If $g = 10 \text{m/s}^2$, the equation of its trajectory is

Q8: A ball projected from ground at an angle of $45^\circ$ just clears a wall in front. If point of projection is $4 m$ from the foot of wall and ball strikes the ground at a distance of $6 m$ on the other side of the wall, the height of the wall is:

Q9: The stream of a river is flowing with a speed of $2 \text{km/h}$. A swimmer can swim at a speed of $4 \text{km/h}$. What should be the direction of the swimmer with respect to the flow of the river to cross the river straight?

Q10: A particle is moving along a circular path with a constant speed of $10 \text{ms}^{-1}$. What is the magnitude of the change in velocity of the particle, when it moves through an angle of $60^\circ$ around the center of circle?

Q11: If a body moving in circular path maintains constant speed of $10 \text{ms}^{-1}$, then which of the following correctly describes relation between acceleration and radius?

Q12: A boat which has a speed of $5 \text{km/hr}$ in still water crosses a river of width $1 \text{km}$ along the shortest possible path in $15 \text{minutes}$. The velocity of the river water in $\text{km/hr}$ is

Q13: A river is flowing from west to east at a speed of $5 \text{meters per minute}$. A man on the south bank of the river, capable of swimming at $10 \text{metres per minute}$ in still water, wants to swim across the river in the shortest time. He should swim in a direction

Q14: Water drops are falling from a nozzle of a shower on to the floor from a height of $9.8 \, m$. Drop falls at a regular interval of time. When the first drop strikes the floor and that is in the third drop begun to fall. Locate the position of the second drop from the floor when the first drop strikes the floor.

Q15: From a tower of height $H$ a particle thrown vertically upward with a speed $U$. The time taken by the particle to hit the ground is $N$ times that to taken to reach the highest point of its path. Find the relation between $H, U$ and $g$.

Q16: A passenger train of length $60 \, m$ travels at a speed $80 \, km/h$. Another freight train of length $120 \, m$ travels at a speed of $30 \, km/h$. Find the ratio of the time taken by the passenger train to completely pass the freight train when they are moving in the same direction and in the opposite direction.

Q17: The position of the particle as function of time is given by $x(t) = At + Bt^2 - Ct^3$. When the particle attain zero acceleration then velocity will be?

Q18: A particle is moving with speed $V = B\sqrt{X}$ along positive $X$ direction calculate the speed of the particle at time $T = \tau$. Assume that the particle is at origin $T=0$.

Q19: A ball is dropped from the top of a $100 \, m$ high tower on a planet. In the last half second before hitting the ground it cover the distance of $19 \, m$. The acceleration due to gravity near the surface of the planet is...

Q20: The engine of a train moving with a uniform acceleration passes a signal post with velocity $U$ and the last component with velocity $V$. The velocity with which the midpoint of the train passes the post is?

Q21: A swimmer wants to cross a river from point A to point B. Line AB makes an angle of $30^\circ$ with the flow of river. Magnitude of velocity of the swimmer is same as that of river. The angle $\theta$ with the line AB should be ____° so that the swimmer reaches point B.

Q22: If the velocity of the body related to displacement $x$ is given as $V = \sqrt{5000 + 24X}$, then the acceleration of the body is $m/s^2$.

Q23: A projectile is given an initial velocity of $(\hat{i} + 2\hat{j}) \, m/s$, where $\hat{i}$ is along the ground and $\hat{j}$ is along the vertical. If $g = 10 \, m/s^2$, the equation of its trajectory is.

Q24: A shell is fired from a fixed artillery gun with the initial speed $U$ such that it hits the target on the ground at a distance $R$ from it. If $T_1$ and $T_2$ are the values of the time taken to hit the target in two possible ways, the product $T_1 T_2$ will be.

Q25: The position of the particle A and B as function of time is given as $r_A = (-3t^2 + 8t + C)\hat{i}$ and $r_B = (10 - 8t^3)\hat{j}$. The velocity of B with respect to A at $t=1s$ is given as $\sqrt{V}$. Find $V$.

Q26: A particle is projected from the ground with velocity $U$ at an angle $\theta$. Find the time after which the velocity of projectile is perpendicular to its initial velocity.

Q27: The trajectory of a projectile in a vertical plane is $Y = AX - BX^2$ where A and B are constants. Find the maximum height attained by the particle and the angle of projection from the horizontal.

Q28: A cricket fielder can throw a cricket ball at a speed $V_0$. If he throw the ball with running with the speed $U$ at an angle $\theta$ to the horizontal. What is the effective angle of horizontal at which the ball is projected in air as seen by the spectator.

Q29: A projectile is fixed at an angle $\theta$ with a horizontal. The condition under which it lands perpendicular on an inclined plane of inclination $\alpha$ is...

Q30: A projectile when the angle of projection is $75^\circ$ a ball falls $10 \, m$ short than the target. When the angle of projection is $45^\circ$ it falls $10 \, m$ ahead of the target. Find the distance of the target from the point of projection $X$.