Motion in a Plane Set-1

Q1: An object is thrown vertically upwards. At its maximum height, which of the following quantity becomes zero?

Q2: A ball is projected vertically upward with an initial velocity of $50 \text{ms}^{-1}$ at $t = 0s$. At $t = 2s$, another ball is projected vertically upward with same velocity. At $t =$ ______ s, second ball will meet the first ball ($g = 10 \text{ms}^{-2}$)

Q3: A body is projected from the ground at an angle of $45^\circ$ with the horizontal. Its velocity after $2s$ is $20 \text{ms}^{-1}$. The maximum height reached by the body during its motion is ______ m. (use $g = 10 \text{ms}^{-2}$)

Q4: A fighter jet is flying horizontally at a certain altitude with a speed of $200 \text{ms}^{-1}$. When it passes directly overhead an anti-aircraft gun, a bullet is fired from the gun, at an angle $\theta$ with the horizontal, to hit the jet. If the bullet speed is $400 \text{m/s}$, the value of $\theta$ will be…….

Q5: A projectile is launched at an angle $\alpha$ with the horizontal with a velocity $20 \text{ms}^{-1}$. After $10 s$, its inclination with horizontal is $\beta$. The value of $\tan\beta$ will be: ($g = 10 \text{ms}^{-2}$)

Q6: A girl standing on road holds her umbrella at $45^\circ$ with the vertical to keep the rain away. If she starts running without umbrella with a speed of $15\sqrt{2} \text{km/h}$, the rain drops hit her head vertically. The speed of rain drops with respect to the moving girl is :

Q7: Motion of particle in $x$–$y$ plane is described by a set of following equations $x = 4 \sin \left( \frac{\pi}{2} - \omega t \right) \text{m}$ and $y = 4 \sin (\omega t) \text{m}$. The path of particle will be

Q8: A person can throw a ball upto a maximum range of $100 m$. How high above the ground he can throw the same ball?

Q9: A particle is moving in a straight line such that its velocity is increasing at $5 \text{ms}^{-1}$ per meter. The acceleration of the particle is ______ $\text{ms}^{-2}$ at a point where its velocity is $20 \text{ms}^{-1}$.

Q10: Two projectiles are thrown with same initial velocity making an angle of $45^\circ$ and $30^\circ$ with the horizontal respectively. The ratio of their range respectively will be

Q11: A ball is thrown vertically upwards with a velocity of $19.6 \text{ms}^{-1}$ from the top of a tower. The ball strikes the ground after $6 s$. The height from the ground up to which the ball can rise will be $\left(\frac{k}{5}\right) m$. The value of $k$ is ______ (use $g = 9.8 \text{m/s}^{-2}$)

Q12: A ball is projected from the ground with a speed $15\text{ms}^{-1}$ at an angle $\theta$ with horizontal so that its range and maximum height are equal, then ‘$\tan \theta$’ will be equal to

Q13: Two projectiles thrown at $30^\circ$ and $45^\circ$ with the horizontal respectively, reach the maximum height in same time. The ratio of their initial velocities is

Q14: If the initial velocity in horizontal direction of a projectile is unit vector $\hat{i}$ and the equation of trajectory is $y = 5x(1 - x)$. The $y$ component vector of the initial velocity is ______ $\hat{j}$. (Take $g = 10\text{m/s}^2$)

Q15: A body of mass $10 kg$ is projected at an angle of $45^\circ$ with the horizontal. The trajectory of the body is observed to pass through a point $(20, 10)$. If $T$ is the time of flight, then the momentum vector, at time $t = \frac{T}{\sqrt{2}}$, is ______ (Take $g = 10 \text{m/s}^2$)

Q16: A ball is projected with kinetic energy $E$, at an angle of $60^\circ$ to the horizontal. The kinetic energy of this ball at the highest point of its flight will become:

Q17: An object is projected in the air with initial velocity $u$ at an angle $\theta$. The projectile motion is such that the horizontal range $R$, is maximum. Another object is projected in the air with a horizontal range half of the range of first object. The initial velocity of the angle of projection, at which the second object is projected, will be ______ degree. (Mark the smallest angle possible)

Q18: A person is swimming with a speed of $10 \text{m/s}$ at an angle of $120^\circ$ with the flow and reaches to a point directly opposite on the other side of the river. The speed of the flow is ‘$x$’ m/s. The value of ‘$x$’ to the nearest integer is ______.

Q19: A force $\vec{F} = (40\hat{i} + 10\hat{j})N$ acts on a body of mass $5 kg$. If the body starts from rest, its position vector $\vec{r}$ at time $t = 10 s$, will be:

Q20: 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 the river. The angle $\theta$ with the line AB should be ______, so that the swimmer reaches point B.

Q21: A bomb is dropped by fighter plane flying horizontally. To an observer sitting in the plane, the trajectory of the bomb is a:

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

Q23: A player kicks a football with an initial speed of $25 \text{ms}^{-1}$ at an angle of $45^\circ$ from the ground. What are the maximum height and the time taken by the football to reach at the highest point during motion? (Take $g = 10 \text{ms}^{-2}$)

Q24: A helicopter is flying horizontally with a speed '$v$' at an altitude '$h$' has to drop a food packet for a man on the ground. What is the distance of helicopter from the man when the food packet is dropped?

Q25: A particle is moving with constant acceleration '$a$'. Following graph shows $v^2$ versus $x$ (displacement) plot. The acceleration of the particle is___$\text{m/s}^2$.

Q26: Starting from the origin at time $t = 0$, with initial velocity $5\hat{j} \text{ms}^{-1}$, a particle moves in the $x$–$y$ plane with a constant acceleration of $(10\hat{i} + 4\hat{j}) \text{ms}^{-2}$. At time $t$, its coordinates are $(20 m, y_0 m)$. The values of $t$ and $y_0$, are respectively:

Q27: A Tennis ball is released from a height $h$ and after freely falling on a wooden floor it rebounds and reaches height $\frac{h}{2}$. The velocity versus height of the ball during its motion may be represented graphically by:

Q28: When a car is at rest, its driver sees rain drops falling on it vertically. When driving the car with speed $v$, he sees that rain drops are coming at an angle $60^\circ$ from the horizontal. On further increasing the speed of the car to $(1 + \beta)v$, this angle changes to $45^\circ$. The value of $\beta$ is close to:

Q29: A particle starts from the origin at $t = 0$ with an initial velocity of $3.0\hat{i} \text{m/s}$ and moves in the $x$–$y$ plane with a constant acceleration $(6.0\hat{i} + 4.0\hat{j}) \text{m/s}^2$. The $x$-coordinate of the particle at the instant when its $y$-coordinate is $32 m$ is $D$ meters. The value of $D$ is:

Q30: A particle moves such that its position vector $\vec{r}(t) = \cos \omega t \hat{i} + \sin \omega t \hat{j}$ where $\omega$ is a constant and $t$ is time. Then which of the following statements is true for the velocity $\vec{v}(t)$ and acceleration $\vec{a}(t)$ of the particle: