Motion in plane practice · Mixed difficulty

If $\theta$ is the angle between the vectors $\mathbf{a}=2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\mathbf{b}=6 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$, then

A particle is projected upwards with a velocity of $100 \mathrm{~m} / \mathrm{sec}$ at an angle of $60^{\circ}$ with the vertical. Find the time in seconds when the particle will move perpendicular to its initial direction, taking ${g}=10 \mathrm{~m} / \mathrm{sec}^2$

If two non-parallel vectors $\vec{A}$ and $\vec{B}$ are equal in magnitude, then the vectors $(\vec{A}-\vec{B})$ and $(\vec{A}+\vec{B})$ will be

A stone is thrown with a velocity $u$ making an angle $\theta \,$ with the horizontal. The horizontal distance covered by its fall to ground is maximum when the angle $\theta \,$ is equal to

Two particles are projected from the same point on ground simultaneously with speed $20 \mathrm{~m} / \mathrm{s}$ and $\frac{20}{\sqrt{3}} \mathrm{~m} / \mathrm{s}$ at angles $30^{\circ}$ and $60^{\circ}$ respectively with the horizontal in the same direction. The maximum distance between them till both of them strike the ground is approximately $({g}=10 \mathrm{~m} / \mathrm{s}^2)$

If a,b and c are unit vectors, then $|\mathbf{a}-\mathbf{b}|^2+|\mathbf{b}-\mathbf{c}|^2+|\mathbf{c}-\mathbf{a}|^2$ does not exceed

Assertion: A man can cross river of width $d$ in minimum time. On increasing the river velocity, minimum time to cross the river by man will remain unchanged.
Reason: As velocity of a river is perpendicular to its width so time to cross the river is independent of velocity of river.

The maximum range of a projectile is $100 \mathrm{~m}$ The maximum height reached by it is

$\mathrm{If}\,(\mathbf{a}+3 \mathbf{b}) \cdot(7 \mathbf{a}-5 \mathbf{b})=0$ and
$(\mathbf{a}-4 \mathbf{b}) \cdot(7 \mathbf{a}-2 \mathbf{b})=0$. Then, the angle between $\mathbf{a}$ and $\mathbf{b}$ is

A ball is projected horizontally from the top of a tower with a velocity $v_0$. It will be moving at an angle of $60^{\circ}$ with the horizontal after time,

At the top of trajectory of a projectile, the acceleration is

The equation of trajectory of a projectile in the vertical plane is $y = \left[ {10x - \left( {\frac{5}{9}} \right){x^2}} \right]m$. the horizontal distance covered by the projectile when it reaches the maximum height is $\left[ {g = 10\,\,m{s^{ - 2}}} \right]$

A particle has two equal accelerations in two given directions. If one of the accelerations is halved, then the angle which the resultant makes with the other is also halved. The angle between the accelerations is

$\overrightarrow A$ is a vector quantity such that $\left| {\overrightarrow A } \right| = \,$ non-zero constant. Which of the following expression is true for $\overrightarrow A$?

Let a, b,c, be vectors of length 3, 4, 5 respectively and $\mathbf{a}$ be perpendicular to $(\mathbf{b}+\mathbf{c}), \mathbf{b}$ to $(\mathbf{c}+\mathbf{a})$ and $\mathbf{c}$ to $(\mathbf{a}+\mathbf{b})$, then the value of $(\mathbf{a}+\mathbf{b}+\mathbf{c})$ is

The lines $\vec{r}=2 \hat{k}+\lambda(3 \hat{i}+2 \hat{j}+\hat{k})$ and $\vec{r}=3 \hat{i}+2 \hat{j}+3 \hat{k}$ $+\mu(6 \hat{i}+4 \hat{j}+2 \hat{k})$ are:

Given that, ${P}+{Q}+{R}=0$. Which of the following statement is true?

Statement-I: Let $\vec{a}=\hat{i}+\hat{j}$ and $\vec{b}=\hat{i}-\hat{j}$ be two vectors. Angle between $\hat{a}+\hat{b}$ and $\hat{a}-\hat{b}=90^{\circ}$
Statement-II: Projection of $\vec{a}+\vec{b}$ on $\vec{a}-\vec{b}$ is zero.

Two forces whose magnitudes are in the ratio 3:5 give a resultant of 35N. If the angle between them is ${60^\mathrm{o}}$, the magnitude of each force is

The vectors $\mathbf{c}, \mathbf{a}=x \hat{\mathbf{i}}+y \hat{\mathbf{j}}+z \hat{\mathbf{k}}$ and $\mathbf{b}=\hat{\mathbf{j}}$ are such that $\mathbf{a}, \mathbf{c}$ and $\mathbf{b}$ form a right handed system, then $\mathbf{c}$ is

Assertion: If $|\vec{A}+\vec{B}|=| \vec{A}-\vec{B} \mid$, then the angle between $\vec{A}$ and $\vec{B}$ is $90^{\circ}$.
Reason: $\vec{A}+\vec{B}=\vec{B}+\vec{A}$

The resultant of two forces at right angle is 5N. When the angle between them is ${120^0}$, the resultant is $\sqrt {13}$. Then the force are

Assertion (A): The distance between two floating objects on a river flowing uniformly does not changes with time.

Reason (R): The floating object has the same velocity as that of the river.

If the angle between two forces increases, the magnitude of their resultant

The sine of the angle between the vector $\mathbf{a}=3 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\mathbf{b}=2 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ is

Two bodies are projected from roof with same speed in different directions. If air resistance is not taken into account then

Assertion: Commutative law is followed while vector addition.
Reason: Addition of two vectors maybe done graphically using head-to-tail method or parallelogram method.

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

What is the range of a projectile thrown with velocity $98 \mathrm{~ms}^{-1}$ with angle $30^{\circ}$ from horizontal ?