Motion in plane practice · Medium

The vector projection of a vector $3\widehat i + 4\widehat k$ on y-axis is

The angle between the z-axis and the vector $\hat i + \hat j + \sqrt 2 \hat k$ is

A steam boat goes across a lake and comes back (i) on a quiet day when the water is still and (ii) on a rough day when there is a uniform current so as to help the journey onwards and to impede the journey back. If the speed of the launch on both days was same, the time required for complete journey on the rough day, as compared to the quiet day will be

Angle between $\mathrm{\vec A}\,and\,\mathrm{\vec B}\,\,is\,\theta$ What is the value of $\mathrm{\vec A}.\left( {\mathrm{\vec B} \times \mathrm{\vec A}} \right)?$

Let $\vec{a}$ and $\vec{b}$ be two vectors inclined at an angle $\theta$, where $\frac{\pi}{2} < \theta < \pi$.
Statement-I: $|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}| < |\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}|$
Statement-II: $|\vec{a}+\vec{b}|^2=a^2+b^2+2\vec{a} \cdot \vec{b}$

The minimum number of vectors having different planes which can be added to give zero resultant is

A projectile can have the same range R for two angles of projection. If ${T_1}$ and ${T_2}$ be the times of flights in the two cases, then the product of the two times of flights is directly proportional to

Two swimmers start swimming perpendicular to river flow with different constant speeds at the same time. After reaching the opposite bank they have to walk back to reach the point exactly opposite to the starting point at the same time. Assume constant walking speed for both the swimmers.

Two vectors $\mathrm{\vec A}\,and\,\mathrm{\vec B}$ are inclined to each other at an angle &theta;. Which of the following is the unit vector perpendicular to both $\mathrm{\vec A}\,and\,\mathrm{\vec B}$ ?

Two particles are projected simultaneously in the same vertical plane, from the same point, both with different speeds and at different angles with horizontal. The path followed by one, as seed by the other, is

$\left( {\mathrm{\vec P} + \mathrm{\vec Q}} \right)$ is a unit vector along X-axis. If $\mathrm{\vec P} = \hat i - \hat j + \hat k,$ then what value is $\mathrm{\vec Q}?$

The magnitudes of a set of 3 vectors are given below. The set of vectors for which the resultant cannot be zero is

A bomb is dropped from an aeroplane moving horizontally at constant speed. When air resistance is taken into consideration, the bomb

Consider the following statements and select the correct statements from the following.
I. Addition and subtraction of scalars make sense only for quantities with same units
II. Multiplication and division of scalars with different units is possible
III. Addition, subtraction, multiplication and division of scalars with same unit is possible

Assertion : A shell fired from a gun is moving along the parabolic path. If it explodes at the top of the trajectory, then no part of the shell can fly vertically.
Reason : The vertical momentum of the shell at the top of the trajectory is zero.

A mosquito is moving with a velocity $V\, = \,(0.5\,{t^2}\hat i\, + \,3\,t\hat j\, + \,9\hat k)\,m/s$ and accelerating in uniform conditions. What will be the directions of mosquito after 2s?

The ranges and heights for two projectiles projected with the same initial velocity at angles ${42^\mathrm{o}}\,\,\mathrm{and }\,\mathrm{4}{\mathrm{8}^\mathrm{o}}$ with the horizontal are ${R_1},{R_2}\mathrm{ }\,\,\mathrm{and }\,{H_1},{H_2},$ respectively. Choose the correct option.

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

At the point of a projectile motion, acceleration and velocity are perpendicular to each other?

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&nbsp;$\sqrt 2$ times that of either of the two having equal magnitude. Then the angles between vectors are given by

Given $\theta$ is the angle between $\mathrm{\vec A}\,and\,\mathrm{\vec B}$. Then $\left| {\hat A \times \hat B\;} \right|$ is equal to

A river flows with a speed more than the maximum speed with which a person can swim in still water. He intends to cross the river by the shortest possible path (i.e., he wants to reach the point on the opposite bank which directly opposite to the starting point). Which of the following is correct?

Given $\mathrm{\vec c} = \mathrm{\vec a} \times \mathrm{\vec b}$. The angle which $\mathrm{\vec a}$ makes with $\mathrm{\vec c}$ is

Statement-I: If $\vec{a}$ is perpendicular to $\vec{b}$ and $\overrightarrow{{c}}$, then $\overrightarrow{{a}} \times(\overrightarrow{{b}} \times \overrightarrow{{c}})=0$
Statement-II: If $\overrightarrow{{b}}$ is perpendicular to $\overrightarrow{{c}}$, then $\overrightarrow{{b}} \times \overrightarrow{{c}}=0$

For a particle projected with speed of 20 m/s from ground, the kinetic energy at top most point of trajectory is $\left( {\frac{1}{4}} \right)th$ of the initial kinetic energy. The angle of projection from horizontal is

The simple sum of two co-initial vectors is 10 units. Their vector sum is 8 units. The resultant of the vectors is perpendicular to the smaller vector. The magnitudes of the two vectors are

If $u$ is the initial velocity of a projectile and $v$ is the velocity at any instant, then the maximum horizontal range $\mathrm{R}_{\mathrm{m}}$ is equal to

One of the rectangular components of a velocity of&nbsp;$60\,\,km{h^{ - 1\,\,\,}}\,\mathrm{is}\,\,\,30\,\,km{h^{ - 1}}$ .The other rectangular component is

Twelve forces each of 5 N act on a body simultaneously. If each force makes an angle of $30^\circ$ with other their resultant is

If $\frac{1}{2}\hat i + \frac{1}{2}\hat j + c\hat k$ is a unit vector, then the value of ${c^2}$ is