Motion in plane practice · Medium

A projectile is fired at an angle of $30^{\circ}$ to the horizontal such that the vertical component of its initial velocity is $80 {~ms}^{-1}$. Its time of flight is $T$. Its velocity at $t=\frac{T}{4}$ has a magnitude of nearly

The trajectory of a projectile is represented by $y=\sqrt{3} x-g x^2 / 2$. The angle of projection is

If $\hat{A}=3 \hat{i}+4 \hat{j}$ and $B=7 \hat{i}+24 \hat{j}$ then a vector having same magnitude as $\vec{B}$ and parallel to vector $\vec{A}$ will be

The angle between $A = i + j$ and $B = i - j$ is

Assertion: The magnitude of the resultant of two vectors $\vec{P}$ and $\vec{Q}$ is maximum $(P+Q)$, when the two vectors act in the same direction and minimum $(P-Q)$, when they act in opposite directions.

Reason: The resultant of two vectors can be found by using the relation

$R=\sqrt{P^2+Q^2+2 P Q \cos \theta}$

Among the following the set that cannot represent the three sides of a triangle is

Statement-I: $\vec{a}=\hat{i}+p \hat{j}+2 \hat{k}$ and $\vec{b}=2 \hat{i}+3 \hat{j}+q \hat{k}$ are parallel vectors if $p=\frac{3}{2}, q=4$.
Statement-II: If $\vec{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}$ and $\vec{b}=b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k}$ are parallel, then $\frac{a_1}{b_1}=\frac{a_2}{b_2}-\frac{a_3}{b_3}$

Which of the following statements are correct ?
I. Centripetal acceleration is always directed towards the centre.
II. Magnitude of the centripetal acceleration is $\frac{{{v^2}}}{R}$.
III. Direction of centripetal acceleration changes pointing always towards the centre.

The resultant of the two vectors $\vec{A}$ and $\vec{B}$ makes angle $\alpha$ with $\vec{A}$ and $\beta$ with $\vec{B}$.

Statement (A) : $\alpha<\beta$ if $|\vec{A}|<|\vec{B}|$

Statement (B) : $\alpha<\boldsymbol{\beta}$ if $|\overrightarrow{{A}}|>|\vec{B}|$

Statement (C) : $\alpha=\beta$ if $|\vec{A}|=|\vec{B}|$

Two vectors are given by $\vec{A}=\hat{i}+2 \hat{j}+2 \hat{k}$ and $\vec{B}=3 \hat{i}+6 \hat{j}+2 \hat{k}$. Another vector $\vec{C}$ has the same magnitude as $\vec{B}$ but has the same direction as $\vec{A}$. Then which of the following vectors represent $\vec{C}$

The angle between two vectors of magnitude 12 and 18 units when their resultant is 24 units, will be:

The vectors from origin to the points $A$ and $B$ are ${A}=3 \hat{{i}}-6 \hat{{j}}+2 \hat{{k}}$ and ${B}=2 \hat{{i}}+\hat{{j}}-2 \hat{{k}}$, respectively. The area of the $\triangle O A B$ be

There are $N$ coplanar vectors each of magnitude $V$. Each vector is inclined to the preceding vector at angle $\frac{2 \pi}{N}$. What is the magnitude of their resultant?

Assertion: Vector $(\hat{i}+\hat{j}+\hat{k})$ is perpendicular to $(\hat{i}-2 \hat{j}+\hat{k})$
Reason: Two non-zero vectors are perpendicular if their dot product is zero.

The friction of the air causes a vertical retardation equal to $10 \%$ of the acceleration due to gravity. (Take $\mathrm{g}=10 \mathrm{~ms}^{-2}$ ). The maximum height will be decreased by in projectile motion -

Statement I : Rain is falling vertically downwards with a velocity of $3 \mathrm{~kmh}^{-1}$. A man walks with a velocity of $4 \mathrm{kmh}^{-1}$. Relative velocity of rain w.r.t. man is $5 \mathrm{~kmh}^{-1}$.

Statement II : Relative velocity of rain w.r.t. man is given by $\quad \mathbf{v}_{m m}=\mathbf{v}_r-\mathbf{v}_m$

What is the component of $3 \hat{i}+4 \hat{j}$ along $\hat{i}+\hat{j}$ ?

Statement-I: The vector product of a force $\vec{F}$ and displacement $\vec{r}$ is equal to the work done.
Statement-II: Work done is not a vector.

Statement-I: If $\theta$ be the angle between $\vec{a}$ and $\overrightarrow{\mathrm{b}}$, then $\tan \theta=\frac{|\vec{a} \times \vec{b}|}{\vec{a} \cdot \vec{b}}$
Statement-II: $\vec{a} \times \vec{b}$ is perpendicular to $\vec{a} . \vec{b}$

Which of the following is not the vector quantity?

Two projectiles are thrown with different velocities and at different angles so as to cover the same maximum height. Show that the sum of the times taken by each to reach the highest point is equal to the total time taken by either of the projectiles.

Which of the following is not true about vectors $\vec{A}, \vec{B}$ and $\vec{C}$ ?

A stone is thrown vertically at a speed of $30 \mathrm{~ms}^{-1}$ taking an angle of $45^{\circ}$ with the horizontal. What is the maximum height reached by the stone? Take $\mathrm{g}=10 \mathrm{~ms}^{-2}$.

Following forces start acting on a particle at rest at the origin of the co-ordinate system simultaneously ${\mathrm{\vec F}_1} = 5\hat i - 5\hat j + 5\hat k,\mathrm{\;\;\;}{\mathrm{F}_2} = 2\hat i + 8\hat j + 6\hat k,{\mathrm{F}_3} = - 6\hat i + 4\hat j - 7\hat k,{\mathrm{F}_4} = - \hat i - 3\hat j - 2\hat k.$ The particle will move

A projectile is projected at an angle of $47^{\circ}$ relative to the horizontal. The range of the projectile could be increased by

A bullet is fired from a cannon with velocity 500 m/s . If the angle of projection is $15^\circ$ and $g = 10\;m/{s^2}$. Then the range is

The vectors $\vec{A}=\hat{i}+2 \hat{j}+3 \hat{k}$ and $\vec{B}=2 \hat{i}-\hat{j}$ are ___ to each other.

The shape of trajectory of the motion of an object is determined by

A particle moves with velocity $3\hat i - 2\hat j + 3\hat k\,\,\,{\text{m}}/{\text{s}}$ under the influence of a constant force $\overrightarrow F = 20\hat i + 15\hat j - 5\hat k\,\,\,{\text{N}}.$ The instantaneous power applied to the particle is (in J/s)

Assertion: In order to hit a target, a man should point his rifle at a point higher than the target.
Reason: The bullet suffers a vertically downward deflection due to gravity.