Motion in plane practice · Mixed difficulty

A boat moves with a speed of 5 km/h relative to water in a river flowing with a speed of 3 km/h and having a width of 1 km. The minimum time taken around a round trip is

A body is projected with velocity ‘$u$’ so that the maximum height is thrice the horizontal range. Then the maximum height is

Two stones are projected from the same speed but making different angels with the horizontal. Their horizontal ranges are equal. The angle of projection of one is $\pi /3$ and the maximum height reached by it is 102 m. Then maximum height reached by the other in metre is

The sum of two vectors $\mathrm{\vec A}\,and\,\mathrm{\vec B}$ is at right angles to their difference. Then

If $a \hat{\mathbf{i}}+b \hat{\mathbf{j}}$ is a unit vector and it is perpendicular to $\hat{\mathbf{i}}+\hat{\mathbf{j}}$, then value of $a$ and $b$ is

Eighteen forces each equal to 10 N act on a particle simultaneously, at an angle $20^{\circ}$ with the next one. If one force is removed, then the resultant of remaining forces is

$\mathrm{\vec A} = 3\hat i - \hat j + 7\hat k\,and\,\mathrm{\vec B} = 5\hat i - \hat j + 9\hat k$ the direction cosine m of the vector $\mathrm{\vec A} + \mathrm{\vec B}$ is

If $\mathrm{\vec A} + \mathrm{\vec B} = \mathrm{\vec C}\,and\,A = \sqrt 3 ,\,B = \sqrt 3 \,and\,C = 3$, then the angle between $\mathrm{\vec A}\;and\;\mathrm{\vec B}$ is

The magnitude of the vectors product o two vectors is $\sqrt 3$ times their scalar product. The angle between the two vectors is

A particle moves in the x-y plane with velocity ${v_x} = 8t - 2$ and ${v_y} = 2$. If it passes through the point  x=14 and y=4 at t=2s, find the equation (x-y relation) of the path

If $\left| {\bar A + \bar B} \right| = \left| {\bar A} \right| = \left| {\bar B} \right|\,$, the angle between $\overrightarrow A$ and $\overrightarrow B$ is

In a projectile motion, velocity at maximum height is

If $\vec a$ and $\vec b$ are two unit vectors such that $\bar a + 2\vec b$ and $5\vec a - 4\vec b$ are perpendicular to each other then the angle between $\vec a$ and $\vec b$ is

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

If a vector $\vec P$ make angles $\alpha ,\,\beta ,$ and $\gamma$ respectively with the $X,\,\,Y$ and $Z$ axes. Then ${\sin ^2}\alpha \,\, + \,\,{\sin ^2}\beta \,\, + \,\,{\sin ^2}\gamma \,\, =$

The component of $\overrightarrow A$ along $\overrightarrow B$ is 2 times that of the component of $\overrightarrow B$ along $\overrightarrow A$. Then A : B is

A bat and an insect are flying with velocities $\overrightarrow{{V}}_{\mathrm{BG}}$ and $\overrightarrow{{V}}_{\text {IG }}$ respectively with respect to ground (in unit vector $\hat{{i}}, \hat{{j}}$ rotation)

$\vec{V}_{\text {BG }}=3.5 \hat{{i}}+9.2 \hat{{j}}, \overrightarrow{\mathrm{V}}_{\mathrm{IG}}=-2.5 \hat{{i}}+{1 . 8} \hat{{j}}$

Calculate the velocity of the insect with respect to bat.

If $\theta$ is the angle between two vectors, then the resultant vector is maximum, when value of $\theta$ is

The vectors $a \hat{i}+2 \hat{j}+6 \hat{k}$ and $2 \hat{i}+2 \hat{j}+6 \hat{k}$ are equal vectors. The value of $a$ is

Given, $\mathrm{\vec P} = \mathrm{\vec A} + \mathrm{\vec B}\,\,\mathrm{and}\,P = A + B.$ The angle between $\mathrm{\vec A}\,and\,\mathrm{\vec B}$ is

Statement-I: If $|\vec{a}+\vec{b}|=|\vec{a}-\vec{b}|$, then angle between $\vec{a}$ and $\vec{b}$ is $90^{\circ}$
Statement-II: $\vec{a}+\vec{b}=\vec{b}+\vec{a}$

Two forces $\overrightarrow {{\mathrm{F}_1}} \;\,and\,\overrightarrow {{\mathrm{F}_2}}$ are acting at right angles to each other. Then their resultant is

The equation of trajectory of projectile is $y = \sqrt 3 x - \frac{g}{2}{x^2}$ the horizontal range for projectile is

The vectors sum of two forces is perpendicular to their vector differences in that case. The forces

If $\mathrm{\vec A} = \mathrm{\vec B},$ then which of the following is not correct

$\mathbf{A}$ 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

A projectile is fired with a velocity v at an angle $\theta$ with the horizontal. The speed of the projectile when its direction of motion makes an angle β with the horizontal is

Apparent dips in two planes normal to each other are $30^{\circ}$ and $45^{\circ}$ respectively. True dip at that place is

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

For vectors $\overrightarrow A \,$ and $\overrightarrow B \,$ making an angle $\theta$ which one of the following relations is correct