If a force $\overset{\rightarrow}{F} = \left(3 \hat{i} + 4 \hat{j} - 5 \hat{k}\right)$ unit acts on a box and the displacement of the box is $\overset{\rightarrow}{d} = \left(5 \hat{i} + 4 \hat{j} + 3 \hat{k}\right)$ unit, the angle between force and displacement is:
1. $\cos^{- 1} \left(0 . 32\right)$

2. $\dfrac{\pi}{2}$

3. $0$

4. $\cos^{- 1} \left(0 . 2\right)$

It is well known that a raindrop falls under the influence of the downward gravitational force and the opposing resistive force. The latter is known to be proportional to the speed of the drop but is otherwise undetermined. Consider a drop of mass $1.00$ g falling from a height of $1.00$ km. It hits the ground with a speed of $50.0$ m/s. Work done by the gravitational force and work done by the unknown resistive force respectively are:

A cyclist comes to a skidding stop in $10~\text m$. During this process, the force on the cycle due to the road is $200~\text N$ and is directly opposed to the motion. Work done by the road on the cycle and work done by the cycle on the road respectively are:
1. $-2000~\text J$ and $2000~\text J$ 
2. $2000~\text J$ and $-2000~\text J$ 
3. $0~\text J$ and $2000~\text J$ 
4. $-2000~\text J$ and $0~\text J$

In a ballistics demonstration, a police officer fires a bullet of mass $50.0~\text g$ with speed $200~\text{m/s}$ on soft plywood of thickness $2.00~\text {cm}.$ The bullet emerges with only $\text{10%}$ of its initial kinetic energy. The emergent speed of the bullet is:

A woman pushes a trunk on a railway platform which has a rough surface. She applies a force of $100$ N over a distance of $10$ m. Thereafter, she gets progressively tired and her applied force reduces linearly with distance to $50$ N. The total distance through which the trunk has been moved is $20$ m. The plot of force applied by the woman and the frictional force, which is $50$ N versus displacement is given below. Work done by the two forces over $20$ m are:

A block of mass $m=1$ kg, moving on a horizontal surface with speed $v_i=\mathrm{2~m/s}$ enters a rough patch ranging from ${x=0.10~\text m}$ to ${x=2.01~\text m}$. The retarding force $F_r$ on the block in this range is inversely proportional to $x$ over this range,

$\begin{aligned} {F}_{r} & =\dfrac{-{k}}{x} \text { for } 0.1<{x}<2.01 {~\text{m}} \\ & =0 \quad ~\text { for } {x}<0.1 \text{ m} \text { and } {x}>2.01 \text{ m} \end{aligned}$

where $k=0.5~\text{J}$. What is the final kinetic energy and speed $v_f$ of the block as it crosses this patch?
1. $5$ J and $1$ m/s 
2. $1$ J and $5$ m/s
3. $0.5$ J and $1$ m/s
4. $0.05$ J and $2$ m/s 

A bob of mass m is suspended by a light string of length $L.$ It is imparted a horizontal velocity $v_0$ at the lowest point $A$ such that it completes a semi-circular trajectory in the vertical plane with the string becoming slack only on reaching the topmost point, the ratio of the kinetic energies $\dfrac{K_B}{K_C}$ at points ${B}$ and ${C}$ is: