Motion in plane practice · Medium

Let $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\overrightarrow{0},|\overrightarrow{\mathrm{a}}|=3,|\overrightarrow{\mathrm{b}}|=4,|\overrightarrow{\mathrm{c}}|=5$
Statement-I: Angle between $\vec{\mathrm{a}}$ and $\vec{\mathrm{b}}$ is $\frac{\pi}{2}$
Statement-II: $|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}|^2=|\overrightarrow{\mathrm{a}}|^2+|\overrightarrow{\mathrm{b}}|^2+2|\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}|$

The X and Y components of vector $\mathrm{\vec A}$ have numerical values 6 and 6 respectively and that of $\left( {\mathrm{\vec A} + \mathrm{\vec B}} \right)$ have numerical values 10 and 9. What is the numerical value of ${\mathrm{\vec B}}$ ?

Three bodies A, B and C projected with the same initial speed with angles of projections $30^{\circ}, 45^{\circ}$ and $60^{\circ}$ respectively. The body/bodies having maximum range

Two balls are projected simultaneously in the same vertical plane from the same point with velocities $V_1$ and $V_2$ with angles $\theta_1$ and $\theta_2$ respectively with the horizontal. If $V_1 \cos \theta_1=$ $V_2 \cos \theta_2$, the path of one ball as seen from the position of other ball is:

The number of components in which a vector can be resolved into is

Assertion: Time of flight of a projectile is equal to twice the time taken to attain the maximum height.

Reason: At the highest point of the trajectory, the projectile comes to rest and then begins to fall vertically downward under the effect of gravity.

What should be the angle between $\Delta \vec{A}$ and $\vec{A}$ so that $|\Delta \vec{A}|=\Delta|\vec{A}|$ ?

Two forces $\vec{F}_1=500 \mathrm{~N}$ due east and $\vec{F}_2=250 \mathrm{~N}$ due north have their common initial point. $\vec{F}_2-\vec{F}_1$ is

Statement-I: In $\Delta \mathrm{abc}, \overrightarrow{\mathrm{ab}}+\overrightarrow{\mathrm{bc}}+\overrightarrow{\mathrm{ca}}=0$
Statements-II: If $\overrightarrow{\mathrm{oa}}=\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{ob}}=\overrightarrow{\mathrm{b}}$, then $\overrightarrow{\mathrm{ab}}=\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}$ (triangle law of addition)

In the projectile motion, if air resistance is ignored, the horizontal motion is at

If $\overrightarrow a = 3\hat i + \hat j - 4\hat k$ and $\overrightarrow b = 6\hat i + 5\hat j - 2\hat k,$ then $\left| {\overrightarrow a \times \overrightarrow b } \right|$ is

For any two vectors $\vec{A}$ and $\vec{B}$, then $(\vec{A} \times \vec{B})^2=$

The linear velocity of a rotating body is given by $\vec v = \vec \omega \times \vec r$ If $\vec \omega = \hat i - 2\hat j + 2\hat k\,\,{\text{and }}\vec r = 4\hat j - 3\hat k,$ then the magnitude of $\vec v$ is

Which of the following is true regarding projectile motion?

If $\mathrm{\vec P} = 2\hat i - 3\hat j + \hat k\,and\,\mathrm{\vec Q} = 3\hat i - 2\hat j,\,then\,\mathrm{\vec P}\mathrm{.\vec Q}$ is

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

Select the correct statements about the football thrown in a parabolic path.
I. At the highest point the vertical component of velocity is zero
II. At the highest point, the velocity of the football, acts horizontally
III. At the highest point, the acceleration of the ball acts vertically downwards

The diagonals of a parallelogram are given by the vectors $3 \hat{i}+\hat{j}+2 \hat{k}$ and $\hat{i}-3 \hat{j}+4 \hat{k}$. Find the area of the parallelogram.

A particle is projected with an initial velocity $\vec{v}=(2 \hat{i}+4 \hat{j}) \mathrm{m} / \mathrm{s}$. The equation of trajectory of the particle will be $\left(g=10 \mathrm{~m} / \mathrm{s}^2\right)$ (Assume horizontal direction as $x$-axis and vertically upward as positive y-axis)

If the magnitudes of two vectors are 3 and 4 and their scalar product is 6, then find the angle between the two vectors.

Assertion: A projectile is thrown with an initial velocity of $(a \hat{i}+b \hat{j}) \mathrm{m} / \mathrm{s}$. If range of projectile is maximum then ${a}={b}$.
Reason: In projectile motion, angle of projection is equal to $45^{\circ}$ for maximum range condition.

A: Two row boats moving parallel to each other and nearby, are attracted towards each other
R: Increase of velocity of fluid flow decrease the pressure

Assertion: Three given concurrent forces may be in equilibrium which is $200 \mathrm{~N} ; 300 \mathrm{~N}$ and 500 $\mathrm{N}$

Reason: Three forces are concurrent and their summation or net force acting on a point is zero when $500 \mathrm{~N}$ acts in one direction and both $200 \mathrm{~N}$ and $300 \mathrm{~N}$ act in the opposite direction.

Two bodies are thrown from the same point with the same velocity of projection, angles of projection $30^\circ$ and $60^\circ .$ If ${R_1}$ and ${R_2}$ are the ranges and ${h_1}$ and ${h_2}$ are maximum heights respectively then

Given $\vec{A}+\vec{B}+\vec{C}+\vec{D}=\overrightarrow{0}$, which of the following statements is not correct?

The maximum and minimum magnitude of the resultant of two given vectors are 17 units and 7 unit respectively. If these two vectors are at right angles to each other, the magnitude of their resultant is

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

For a projectile if $\alpha$ is the angle of projection, $R$ is the range, $h$ is the maximum height, $T$ is the time of flight then

Two shells are fired from a cannon with same speed at angle $\alpha$ and $\beta$ respectively with the horizontal. The time interval between the shots is T. They collide in mid-air after time ' $t$ ' from the first shot. Which of the following conditions must be satisfied?

A swimmer wishes to cross a $800 \mathrm{~m}$ river flowing at $4 \mathrm{~km} / \mathrm{h}$. His speed with respect to water is $3 \mathrm{~km} / \mathrm{h}$. The shortest possible time to cross the river is