Motion in plane practice · Mixed difficulty

As a rocket is accelerating vertically upwards at $9.8 \mathrm{~ms}^{-2}$ near the earth's surface, it releases a projectile with zero speed relative to rocket. Immediately after release, the acceleration (in $\mathrm{ms}^{-2}$ ) of the projectile is [take, $g=9.8 \mathrm{~ms}^{-2}$ ]

A particle is projected in XY plane with Y-axis along vertical, the point of projection is origin. The equation of the path is y= $\sqrt 3 \times - \frac{g}{2}{ \times ^2}$. Where X and Y are in metres. The speed of
projection in $m{s^{ - 1}}$ is

A force of $8 {~N}$ makes an angle $30^{\circ}$ with $X$-axis. Find the $x$ and $y$ components of the force.

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

A bomb is dropped by fighter plane flying horizontally. To an observer sitting in the plane, the trajectory of the bomb is a

Find a vector whose length is 7 and which is perpendicular to each of the vectors: $\vec{A}=2 \hat{i}-3 \hat{j}+6 \hat{k} \text { and } \vec{B}=\hat{i}+\hat{j}-\hat{k}$

The resultant of two forces acting at an angle of $150^{\circ}$ is $10 {~N}$ and is perpendicular to one of the forces. One of the two other forces is

Four bodies $P,\,Q,\,R$ and $S$ are projected with equal velocities having angles of projection $15^\circ ,\,30^\circ ,\,45^\circ$ and $60^\circ$ with the horizontal respectively. The body having shortest range is

A ball is projected horizontally from the top of a building 80 m high. If the line joining the point of projection to the point where it hits the ground makes an angle of $45^{\circ}$ to the horizontal, the initial velocity of the ball is

Assertion: A projectile that traverses a parabolic path shows deviation from its idealized trajectory in the presence of air resistance.

Reason: Air resistance affects the motion of the projectile.

If a,b and c are unit vectors, then $|\mathbf{a}-\mathbf{b}|^2+|\mathbf{b}-\mathbf{c}|^2+|\mathbf{c}-\mathbf{a}|^2$ does not exceed

The magnitude of $\hat{i}+\hat{{j}}$ is

The equation of trajectory of an oblique projectile is $y=x-\frac{1}{2} x^2$. The time of flight of projectile will be.

Statement-I: If $\vec{a} \cdot \vec{b}=\vec{b} \cdot \vec{c}$, then $\vec{a}$ may not always be equal to $\vec{{c}}$.
Statement-II: The dot product of two vectors involves cosine of the angle between the two vectors.

Statement-I: If dot product and cross product of $\vec{a}$ and $\vec{b}$ are zero, it implies that one of the vector $\vec{a}$ and $\vec{b}$ must be a null vector.
Statement-II: Null vector is a vector with zero magnitude.

Two boys $P$ and $Q$ are playing on a river bank. $P$ plans to swim across the river directly and come back. $Q$ plans to swim downstream by a length equal to the width of the river and then come back. Both of them bet each other, claiming that the boy succeeding in less time will win. Assuming the swimming rate of both $P$ and $Q$ to the same, it can be concluded that

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?

If $\vec{a}+\vec{b}$ is perpendicular to $\vec{a}$ and $2 \vec{a}+\vec{b}$ is perpendicular to $\vec{b}$. Then $\frac{|\vec{b}|}{|\vec{a}|}$ is

The position of a particle is given by $\vec{r}=3.0 t \hat{i}+2.0 t^2 \hat{j}+5.0 \hat{k}$ where $t$ is in seconds and the coefficients have the proper units for $r$ to be in metres. Find $v(t)$ and $a(t)$ of the particle.

If the angle between the vectors $\vec{A}$ and $\vec{B}$ is $\theta$, the value of the product $(\overrightarrow{{B}} \times \overrightarrow{{A}}) \cdot \overrightarrow{{A}}$ is equal:

A body is projected with speed $v\;\mathrm{m}{\mathrm{s}^{ - 1}}$ at angle $\theta$. The kinetic energy at the highest point is half of the initial kinetic energy. The value of $\theta$ is

A large number of bullets are fired in all directions with same speed v. What is the maximum area on the ground on which these bullets will spread

Three balls are projected from the top of a building with equal speeds but at different angles. Balls strike the ground with velocities ${v} 1$, $v2$ and ${v} 3$ respectively, then

Of the vectors given below, the parallel vectors are,

$\begin{aligned} & \vec{A}=6 \hat{i}+8 \hat{j} \\ & \vec{B}=210 \hat{i}+280 \hat{k} \\ & \vec{C}=5.1 \hat{i}+6.8 \hat{j} \\ & \vec{D}=3.6 \hat{i}+6 \hat{j}+48 \hat{k} \end{aligned}$

If $a^2+b^2+c^2=1$ where $a, b, c \in R$, then the maximum value of $(4 a-3 b)^2+(5 b-4 c)^2+(3 c-5 a)^2$ 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 $\overrightarrow{{b}} \times \overrightarrow{{c}}=0$

Two objects are projected with same velocity ' $u$ ' however at different angles $\alpha$ and $\beta$ with the horizontal. If $\alpha+\beta=90^{\circ}$, the ratio of horizontal range of the first object to the 2nd object will be :

A football is kicked into the air vertically upwards with velocity $u.$ The velocity of the ball at the highest point is

If vectors $\hat{i}-3 \hat{j}+5 \hat{k}$ and $\hat{i}-3 \hat{j}-a \hat{k}$ are equal vectors, then the value of $a$ is

Assertion: Angle between $\hat{i}+\hat{j}$ and $\hat{i}$ is $45^{\circ}$.
Reason: $\hat{i}+\hat{j}$ is equally inclined to both $\hat{i}$ and $\hat{j}$ the angle between $\hat{i}$ and $\hat{j}$ is $90^{\circ}$.