Motion in plane practice · Medium

The angle between two vectors $A$ and $B$ is $\theta$. Vector $R$ is the resultant of the two vectors. If $R$ makes an $\frac{\theta}{2}$ with $A$, then

The kinetic energy of a projectile becomes half of its initial kinetic energy at maximum height. The maximum height of the projectile is ($u \to$initial velocity)

The resultant of two vectors of magnitudes 2A and ${\sqrt 2 A\;}$ acting at an angle $\theta$ is $\sqrt {10} A$. The correct value of $\theta$ is

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.

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

Two vectors are given by $\overrightarrow a = - 2\widehat i + \widehat j - 3\widehat k$ and $\overrightarrow b = 5\widehat i + 3\widehat j - 2\widehat k$. If $3\overrightarrow a + 2\overrightarrow b - \overrightarrow C = 0$ then third vector $\overrightarrow c$ is

A cannon on a level plane is aimed at an angle $\theta$ above the horizontal and a shell is fired with muzzle velocity ${v_0}$ towards a cliff D distance away. The height at which the cannon strikes the cliff is given by

Two forces are such that the sum of their magnitudes is 18N and their resultant is perpendicular to the smaller force and magnitude of the resultant is 12N . Then the magnitudes of the forces are

Statement-I: The sum and difference of two vectors will be equal in magnitude, when the two vectors are perpendicular to each other.
Statement-II: If either of the two vectors is a null vector.

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 $|\vec{A}|=|\vec{B}|$

Assertion : When the range of projectile is maximum, the time of flight is the largest.
Reason : Range is maximum when angle of projection is $45^\circ$

Velocity is

What is the angle between $\hat i + \hat j + \hat k\,and\,\hat i$

If $\mathbf{a} \times \mathbf{b}=\mathbf{b} \times \mathbf{c} \neq 0$, then the correct statement is

In the entire path of a projectile, the quantity that remains unchanged is

Three vectors $\vec{a}, \vec{b}$ and $\vec{c}$ satisfy the relation $\vec{a} \cdot \vec{b}=0$ and $\vec{a} \cdot \vec{c}=0$. The vector $\vec{a}$ is parallel to

Two forces of magnitude F have a resultant of the same magnitude F. The angle between the two forces is

Assertion : Two balls of different masses are thrown vertically upward with same speed. They will pass through their point of projection in the downward direction with the same speed.
Reason : The maximum height and downward velocity attained at the point of projection are independent of the mass of the ball.

The horizontal range of a projectile is maximum when the angle of projection is

The angle between $\mathrm{\vec A}\;\,and\,\mathrm{\vec B}\,is\,\theta$, the value of the triple product $\mathrm{\vec A}\mathrm{.\vec B} \times \mathrm{\vec A}$ is

Statement-I: If the magnitude of the sum of two unit vectors is a unit vector, then magnitude of their difference is $\sqrt{3}$
Statement-II: $|\vec{a}|+|\vec{b}|=|\vec{a}+\vec{b}|$

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

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

A boat can travel at a speed of 8 kmph in still water on a lake. In the flowing water of a stream, it can move at 8kmph relative to the water in the stream. If the stream speed is 3kmph, how fast can the boat move past a tree on the shore in travelling (i) upstream (ii) downstream?

Assertion: The time taken by a boat to cover a certain distance upstream is more than the time taken to cover the same distance downstream.

Reason: The speed of the boat is lesser while moving upstream as compared to downstream due to the speed of the stream.

Of the vectors given below, the parallel vectors are,

$\begin{aligned} & \vec{A}=6 \hat{i}+8 \hat{j} \\ & \vec{B}=210 \hat{i}+280 \hat{k} \\ & \vec{C}=5.1 \hat{i}+6.8 \hat{j} \\ & \vec{D}=3.6 \hat{i}+6 \hat{j}+48 \hat{k} \end{aligned}$

Given$\mathrm{\vec P}\mathrm{.\vec Q} = 0\,then\,\left| {\mathrm{\vec P} \times \mathrm{\vec Q}} \right|\,is$

The resultant of two vector of equal magnitude is twice of magnitude of either of the vectors the angle between them is

The adjacent sides of a parallelogram are represented by co-initial vectors $2\hat i + 3\hat j\,and\,\hat i + 4\hat j$. The area of the parallelogram is

If the vector $\mathrm{\vec A} = 2\hat i + 4\hat j\,\,and\,\,\mathrm{\vec B} = 5\hat i + \mathrm{p\hat j}$ are parallel to each other, the magnitude of $\mathrm{\vec B}$ is

If $\overrightarrow A = 2\hat i + 3\hat j + 4\hat k\,\,and\,\,\mathrm{\vec B} = 4\hat i + 3\hat j + 2\hat k$, then angle between $\overrightarrow A$ and $\mathrm{\vec B}$ is