Motion in plane practice · Mixed difficulty

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

If air resistance is not considered in projectiles, the horizontal motion takes place with

Assertion(A): When $\theta=45^{\circ}$ or $135^{\circ}$, the value of $R$ remains the same, only the sign changes.

Reason(R): $R=\frac{u^2 \sin 2 \theta}{g}$

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

A body is thrown at an angle $30^\circ$ to the horizontal with the velocity of 30 m/s. After 1 sec its velocity will be (in m/s) $\left( {g = 10\,m/{s^2}} \right).$

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.

I.Inprojectilemotion,theanglebetwee instantaneousvelocityvectorandacceleration vector can be anything between 0 to $\pi$ (excluding the limiting case.)

II. In projectile motion, acceleration vector is always pointing vertically downwards. (Neglect air friction).

Which of the following statement(s) is/are correct?

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

A motorboat covers a given distance in 6 hours moving downstream on a river. It covers the same distance in 10 hours moving upstream. The time it takes to cover the same distance in still water is (in hours).

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

If the resultant of the vector $3\bar i + 4\bar j + 5\bar k$ and $5\bar i + 3\bar j + 4\bar k$ makes an angle $\theta \,$ with x-axis. Then find $\cos \theta$

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

A scalar quantity is one that

Which one of the following pair cannot be the rectangular components of force vector of 10 N?

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

The horizontal and vertical components of a force are 3 N and 4 N respectively. Then force is

If $\mathbf{A} \cdot \mathbf{B}=\mathbf{A} \times \mathbf{B}$, then angle between $\mathbf{A}$ and $\mathbf{B}$ 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: $|\vec{a}|+|\vec{b}|=|\vec{a}+\vec{b}|$