Motion in plane practice · Mixed difficulty

For a projectile thrown into space with a speed v, the horizontal range is $\frac{{\sqrt 3 {v^2}}}{{2\mathrm{g}}}.$ The vertical range is $\frac{{{v^2}}}{{8\mathrm{g}}}$. The angle which the projectile makes with the horizontal initially is

If $|{A} \times {B}|=\sqrt{3} {A} \cdot {B}$, then the value of $|{A}+{B}|$ is

The two vectors have magnitudes 3 and 5 . If angle between them is $60^{\circ}$, then the dot product of two vectors will be :-

Given that $\mathrm{\vec A} + \mathrm{\vec B} + \mathrm{\vec C} = 0$. Out of three vectors, two are equal in magnitude and the magnitude of third vector is $\sqrt 2$ times that of either of the two having equal magnitude. Then the angles between vectors are given by

The unit vector in ZOX plane making angles $45^{\circ}$ and $60^{\circ}$ respectively, with $a=2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\mathbf{b}=\hat{\mathbf{j}}-\hat{\mathbf{k}}$, is

$(\vec{A}-\vec{B}) \times(\vec{A}+\vec{B})=$

A particle is projected with speed $20 \mathrm{~ms}^{-1}$ at an angle $30^{\circ}$ with horizontal. After how much time the angle between velocity and acceleration will be $90^{\circ}$

Which of the following sets of vectors given below may have zero resultant?

If $|\mathbf{a}|=3,|\mathbf{b}|=4$, then a value of $\lambda$ for which $\mathbf{a}+\lambda \mathbf{b}$ is perpendicular to $\mathbf{a}-\lambda \mathbf{b}$.

The range $R$ of projectile is same when its maximum heights are $h_1$ and $h_2$. What is the relation between $R$ and $h_1$ and $h_2$

Assertion: The trajectory of an object moving under constant acceleration due to gravity can be a straight line or parabola.
Reason: Initial conditions affect the motion of the object.

I. Pressure
II. Temperature
III. Momentum
IV. Work
Which of the following physical quantities are scalars?

From the top of a tower, a ball A is dropped and another ball B is thrown horizontally at the same time. Which ball strikes the ground first?

Assertion At highest point of a projectile, dot product of velocity and acceleration is zero.

Reason At highest point, velocity and acceleration are mutually perpendicular.

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

Statement I : In order to hit a target, a man should point his rifle in the same direction as the target.

Statement II : The horizontal range of bullet is dependent on the angle of projection with the horizontal.

If $\overrightarrow {{a_1}}$ and $\overrightarrow {{a_2}}$ are two non collinear unit vectors inclined at ${60^0}$ to each other then the value of $\left( {\overrightarrow {{a_1}} - \overrightarrow {{a_2}} } \right).\left( {\overrightarrow {2{a_1}} + \overrightarrow {{a_2}} } \right)$

The resultant of forces $\overrightarrow {{F_1}} = 4\hat i - 3\hat j$ and $\overrightarrow {{F_2}} = 6\hat i + 8\hat j\,$ is

Assertion: The maximum possible height attained by the projected body is $\frac{{{u^2}}}{{2g}}$ where $u$ is the velocity of projection
Reason : To attain the maximum height, body is thrown vertically upwards.

A bomb is dropped from an aeroplane moving horizontally at constant speed. If air resistance is neglected, then the bomb

A projectile has initially the same horizontal velocity as it would acquire if it had moved from rest with uniform acceleration of $3 \mathrm{~m} \mathrm{s}^{-2}$ for 0.5 minutes. If the maximum height reached by it is $80 \mathrm{~m}$, then the angle of projection is
(Take $g=10 \mathrm{~m} \mathrm{s}^{-2}$ )

If $O A B C$ is a tetrahedron such that $\mathrm{OA}^2+\mathrm{BC}^2$ $=\mathrm{OB}^2+\mathrm{CA}^2=\mathrm{OC}^2+\mathrm{AB}^2$, then which of the following is not true?

Two projectiles A and B are thrown with the same speed such that A makes angle $\mathrm{q}$ with the horizontal and B makes angle $q$ with the vertical, then

Consider the following statements about three vectors $\overrightarrow{{a}}, \overrightarrow{{b}}$ and $\overrightarrow{{c}}$ that have non-zero magnitudes. It follows $\overrightarrow{{b}}=\overrightarrow{{c}}$

(i) if $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}$

(ii) if $\vec{a} \times \vec{b}=\vec{a} \times \vec{c} \quad \vec{b}$ must be perpendicular to $\overrightarrow{\mathrm{c}}$

Statement-I: Two bodies thrown with same speed from the same point at the same instants but at different angles never collide in air.
Statement-II: $x$ and $y$ co-ordinates of the two projectiles always differ

Statement-I: Minimum number of non-equal vectors in a plane required to give zero resultant is three.
Statement-II: If $\vec{a}+\vec{b}+\vec{c}=0$, then they must lie in one plane

Assertion: Two projectiles of masses $m$ and $4m$ when projected with same initial velocity vector have different ranges.
Reason: The horizontal range of a projectile depends on mass of the body.

The component of a vector r along X-axis will
have a maximum value, if :

A body constrained to move along the $z$-axis of a co-ordinate system is subjected to a constant force $\vec{F}$ given by $\vec{F}=-\hat{i}+2 \hat{j}+3 \hat{k}$ newton where $\hat{i}, \hat{j}$ and $\hat{k}$ represent unit vectors along $x,y$ and $z$-axis of the system respectively. Calculate the work done by the force in displacing the body through a distance of $4 \mathrm{~m}$ along the $z$-axis.

$\mathbf{A}$ and $\mathbf{B}$ are two vectors given by $\mathbf{A}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}$ and $\mathbf{B}=2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}$. The magnitude of the component of $\mathbf{A}$ along $\mathbf{B}$ is