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?