Water droplets are coming from an open tap at a particular rate. The spacing between a droplet observed at $4^{\text{th}}$ second after its fall to the next droplet is 34.3 m. At what rate the droplets are coming from the tap? (Take $g = 9.8 \mathrm{~m~s}^{-2}$) (1) 3 drops$/2$ seconds (2) 2 drops$/$ second (3) 1 drop/second (4) 1 drop$/7$ seconds
Two billiard balls of equal mass 30 g strike a rigid wall with same speed of 108 kmph (as shown) but at different angles. If the balls get reflected with the same speed, then the ratio of the magnitude of impulses imparted to ball $a$ and ball $b$ by the wall along $X$ direction is: (1) $1 : 1$ (2) $\sqrt{2} : 1$ (3) $2 : 1$ (4) $1 : \sqrt{2}$
A body of mass 2 kg moving with a speed of $4\mathrm{~m~s}^{-1}$ makes an elastic collision with another body at rest and continues to move in the original direction but with one fourth of its initial speed. The speed of the two body centre of mass is $\frac{x}{10}$ m/s. Find the value of $x$.
A particle of mass $m$ is moving in time $t$ on a trajectory given by, $$\vec{r} = 10\alpha t^{2}\hat{\mathrm{i}} + 5\beta(t - 5)\hat{\mathrm{j}}$$ where $\alpha$ and $\beta$ are dimensional constants. The angular momentum of the particle becomes the same as it was for $t = 0$ at time $t =$ \_\_\_\_ seconds.
In the reported figure, two bodies $A$ and $B$ of masses 200 g and 800 g are attached with the system of springs. Springs are kept in a stretched position with some extension when the system is released. The horizontal surface is assumed to be frictionless. The angular frequency will be \_\_\_\_ rad $\mathrm{s}^{-1}$ when $k = 20\mathrm{~N~m}^{-1}$.
A pendulum bob has a speed of $3\mathrm{~m~s}^{-1}$ at its lowest position. The pendulum is 50 cm long. The speed of bob, when the length makes an angle of $60^{\circ}$ to the vertical will be ($g = 10\mathrm{~m~s}^{-2}$) \_\_\_\_ $\mathrm{m~s}^{-1}$.
A particle of mass 1 mg and charge $q$ is lying at the mid-point of two stationary particles kept at a distance 2 m when each is carrying same charge $q$. If the free charged particle is displaced from its equilibrium position through distance $x$ ($x \ll 1\mathrm{~m}$), the particle executes SHM. Its angular frequency of oscillation will be \_\_\_\_ $\times 10^{5}\mathrm{~rad~s}^{-1}$ (if $q^{2} = 10\mathrm{~C}^{2}$).
If $f: R \rightarrow R$ is a function defined by $f(x) = e^{|x|} - e^{-x}$ / $e^{x} + e^{-x}$, then $f$ is: (1) bijective (2) $f$ is monotonically increasing on $(0, \infty)$ (3) $f$ is monotonically decreasing on $(0, \infty)$ (4) not differentiable at $x = 0$