# For any arbitrary motion in space, which of the following relations is true? 1. $$\vec{v}_{\text {avg }}=\frac{1}{2}\left[\vec{v}\left(t_1\right)+\vec{v}\left(t_2\right)\right]$$ 2. $$\vec{v}(t)=\vec{v}(0)+\vec{a} t$$ 3. $$\vec{r}({t})=\vec{r}(0)+\vec{v}(0){t}+\frac{1}{2} \vec{a}{t}^2$$ 4. $$\vec{v}_{\text {avg }}=\frac{\left[\vec{r}\left(t_2\right)-\vec{r}\left(t_1\right)\right]}{\left(t_2-t_1\right)}$$

Subtopic:  Speed & Velocity |
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A particle is moving along a circle such that it completes one revolution in $$40$$ seconds. In $$2$$ minutes $$20$$ seconds, the ratio of $$|displacement| \over distance$$ will be:
1. $$0$$
2. $$\frac{1}{7}$$
3. $$\frac{2}{7}$$
4. $$\frac{1}{11}$$

Subtopic:  Position & Displacement |
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Consider the motion of the tip of the second hand of a clock. In one minute (assuming $$R$$ to be the length of the second hand), its:

 1 displacement is $$2\pi R$$ 2 distance covered is $$2R$$ 3 displacement is zero. 4 distance covered is zero.
Subtopic:  Position & Displacement |
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A person, reaches a point directly opposite on the other bank of a flowing river, while swimming at a speed of $$5$$ m/s at an angle of $$120^\circ$$ with the flow. The speed of the flow must be:
1. $$2.5$$ m/s
2. $$3$$ m/s
3. $$4$$ m/s
4. $$1.5$$ m/s

Subtopic:  Relative Motion |
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A car with a vertical windshield moves in a rain storm at a speed of $$40$$ km/hr. The rain drops fall vertically with a constant speed of $$20$$ m/s. The angle at which raindrops strike the windshield is:
1. $$\tan^{- 1} \frac{5}{9}$$
2. $$\tan^{- 1} \frac{9}{5}$$
3. $$\tan^{- 1} \frac{3}{2}$$
4. $$\tan^{- 1} \frac{2}{3}$$

Subtopic:  Relative Motion |
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A particle projected from origin moves in the $$x\text-y$$ plane with a velocity $$\overrightarrow{v} = 3 \hat{i} + 6 x \hat{j}$$$\stackrel{}{\mathrm{}}$, where $$\hat i$$ and $$\hat j$$ are the unit vectors along the $$x$$ and $$y\text-$$axis. The equation of path followed by the particle is:
1. $$y=x^2$$
2. $$y=\frac{1}{x^2}$$
3. $$y=2x^2$$
4. $$y=\frac{1}{x}$$

Subtopic:  Speed & Velocity |
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The position coordinates of a projectile projected from ground on a certain planet (with no atmosphere) are given by
$$y =4 t - 2 t^{2}~ \text{m}$$ and $$x =3t$$ metre, where $$t$$ is in seconds and point of projection is taken as the origin. The angle of projection of projectile with vertical is:
1. $$30^{\circ}$$
2. $$37^{\circ}$$
3. $$45^{\circ}$$
4. $$60^{\circ}$$

Subtopic:  Projectile Motion |
78%
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The velocity at the maximum height of a projectile is $$\frac{\sqrt{3}}{2}$$ times its initial velocity of projection $$(u)$$. Its range on the horizontal plane is:
1. $$\frac{\sqrt{3} u^{2}}{2 g}$$
2. $$\frac{3 u^{2}}{2 g}$$
3. $$\frac{3 u^{2}}{ g}$$
4. $$\frac{u^{2}}{2 g}$$

Subtopic:  Projectile Motion |
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The equation of a projectile is $$y = ax -bx^{2}$$. Its horizontal range is?
1. $$\frac{a}{b}$$
2. $$\frac{b}{a}$$
3. $$a+b$$
4. $$b-a$$

Subtopic:  Projectile Motion |
72%
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When a particle is projected at some angle to the horizontal, it has a range $$R$$ and time of flight $$t_1$$. If the same particle is projected with the same speed at some other angle to have the same range, its time of flight is $$t_2$$, then:
1. $$t_{1} + t_{2} = \frac{2 R}{g}$$
2. $$t_{1} - t_{2} = \frac{R}{g}$$
3. $$t_{1} t_{2} = \frac{2 R}{g}$$
4. $$t_{1} t_{2} = \frac{R}{g}$$

Subtopic:  Projectile Motion |
74%
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