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 $g$.