Kinematics

$\vec A$ is a vector quantity such that $|\vec A|=$ non-zero constant. Which of the following expression is true for $\vec A$ ?

Which of the following relations is true for two unit vector $\hat A$ and $\hat B$ making an angle $\theta$ to each other ?

If force $\vec F=3\hat i+4\hat j-2\hat k$ acts on a particle having position vector $2\hat i+\hat j+2\hat k$ then, the torque about the origin will be :

A force on an object of mass 100 g is $\left(10\hat i+5\hat j\right)$ N. The position of that object at t = 2 s is $\left(a\hat i+b\hat j\right)$ m after starting from rest. The value of $\dfrac ab$ will be ______.

Given below are two statements. One is labelled as Assertion A and the other is labelled as Reason R.

Assertion A: Two identical balls $A$ and $B$ thrown with same velocity '$u$' at two different angles with horizontal attained the same range $R$. If $A$ and $B$ reached the maximum height $h_1$ and $h_2$ respectively, then $R=4\sqrt{h_1h_2}$

Reason R: Product of said heights.
$$h_1h_2=\left(\frac{u^2\sin^2\theta}{2g}\right)\cdot\left(\frac{u^2\cos^2\theta}{2g}\right)$$
Choose the correct answer :

Two buses $P$ and $Q$ start from a point at the same time and move in a straight line and their positions are represented by $X_P(t)=\alpha t+\beta t^2$ and $X_Q(t)=ft-t^2$. At what time, both the buses have same velocity ?

For a particle in uniform circular motion, the acceleration $\vec a$ at any point $P(R,\theta)$ on the circular path of radius $R$ is (when $\theta$ is measured from the positive $x$-axis and $v$ is uniform speed) :

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

A fighter jet is flying horizontally at a certain altitude with a speed of 200 m s$^{-1}$. 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 m/s, the value of $\theta$ will be ______°.

A ball of mass 0.5 kg is dropped from the height of 10 m. The height, at which the magnitude of velocity becomes equal to the magnitude of acceleration due to gravity, is ______ m.

[Use $g=10$ m/s$^2$]

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

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

A girl standing on road holds her umbrella at 45° with the vertical to keep the rain away. If she starts running without umbrella with a speed of $15\sqrt2$ kmh$^{-1}$, the rain drops hit her head vertically. The speed of rain drops with respect to the moving girl is :

When a ball is dropped into a lake from a height 4.9 m above the water level, it hits the water with a velocity $v$ and then sinks to the bottom with the constant velocity $v$. It reaches the bottom of the lake 4.0 s after it is dropped. The approximate depth of the lake is :

A particle of mass $m$ is moving in a circular path of constant radius $r$ such that its centripetal acceleration ($a$) is varying with time $t$ as $a=k^2rt^2$, where $k$ is a constant. The power delivered to the particle by the force acting on it is given as

Motion of a particle in $x$-$y$ plane is described by a set of following equations $x=4\sin\left(\dfrac\pi2-\omega t\right)$ m and $y=4\sin(\omega t)$ m. The path of the particle will be :

The position vector of 1 kg object is $\vec r=\left(3\hat i-\hat j\right)$ m and its velocity $\vec v=\left(3\hat j+\hat k\right)$ ms$^{-1}$. The magnitude of its angular momentum is $\sqrt x$ Nm where $x$ is ______.

A ball is spun with angular acceleration $\alpha=6t^2-2t$ where $t$ is in second and $\alpha$ is in rads$^{-2}$. At $t=0$, the ball has angular velocity of 10 rads$^{-1}$ and angular position of 4 rad. The most appropriate expression for the angular position of the ball is :

A car covers AB distance with first one-third at velocity $v_1$ ms$^{-1}$, second one-third at $v_2$ ms$^{-1}$ and last one-third at $v_3$ ms$^{-1}$. If $v_3=3v_1$, $v_2=2v_1$ and $v_1=11$ ms$^{-1}$ then the average velocity of the car is ______ ms$^{-1}$.

Two balls A and B are placed at the top of 180 m tall tower. Ball A is released from the top at $t=0$ s. Ball B is thrown vertically down with an initial velocity '$u$' at $t=2$ s. After a certain time, both balls meet 100 m above the ground. Find the value of '$u$' in ms$^{-1}$. [use $g=10$ ms$^{-2}$] :