Motion in plane practice · Mixed difficulty

Given, vector, ${A}=\hat{{i}}-\hat{{j}}+2 \hat{{k}}$ and vector ${B}=3 \hat{{i}}-3 \hat{{j}}+6 \hat{{k}}$, then which one of the following statements is true?

Two vectors $\mathrm{\vec A}\,and\,\mathrm{\vec B}$ are inclined at an angle $\theta$. Now if the vectors are interchanged then the resultant turns through an angle $\beta$. Which of the following relation is true

A bird moves with velocity $20 \mathrm{~ms}^{-1}$ in a direction making an angle of $60^{\circ}$ with the eastern line and $60^{\circ}$ with vertical upward. Represent the velocity vector in rectangular form.

The component of a vector along any other direction is

For a projection, ${\left( {\mathrm{range}} \right)^2}$ is 48 times of (maximum $\mathrm{height}{)^2}\;$ obtained. Find angle projection.

If three vectors along coordinate axes represent the adjacent sides of a cube of length $b$, then the unit vector along its diagonal passing through the origin will be

In cartesian coordinates the point $A$ is $\left(x_1, y_1\right)$, where $x_1=1$ on the curve $y=x^2+x+10$. Then tangent at $A$ cuts the $X$-axis at $B$. The value of the dot product $\mathbf{O A} \cdot \mathbf{A B}$ is

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

The path followed by a body projected along $y$-axis is given as by $y=\sqrt{3} x-(1 / 2) x^2$, if $g=10 \mathrm{~m} / \mathrm{s}$, then the initial velocity of projectile will be $-(x$ and $y$ are in $m$ )

If $\mathbf{A} \cdot \mathbf{B}=0$ and $\mathbf{A} \times \mathbf{B}=1$, then $A$ and $B$ are

For what value of a,$\mathrm{\vec A} = 2\hat i + \mathrm{a\hat j} + \hat k$ will be perpendicular to $\mathrm{\vec B} = 4\hat i - 2\hat j - \hat k$

A motor boat covers a given distance in 6 h moving down stream of a river. It covers the same distance in 10 h moving upstream. The time (in hour) it takes to cover the same distance in still water is

The greatest height to which a man can throw a stone is $h$. The greatest distance to which he can throw it will be

The maximum height attained by a projectile is increased by $10 \%$. Keeping the angle of projection constant, what is the percentage increase in the time of flight?

In case of a projectile motion, what is the angle between the velocity and acceleration at the highest point

$\mathrm{If}\,(\mathbf{a}+3 \mathbf{b}) \cdot(7 \mathbf{a}-5 \mathbf{b})=0$ and
$(\mathbf{a}-4 \mathbf{b}) \cdot(7 \mathbf{a}-2 \mathbf{b})=0$. Then, the angle between $\mathbf{a}$ and $\mathbf{b}$ is

A bullet is fired with a velocity u making an angle of ${60^0}$ with the horizontal plane. The horizontal component of the velocity of the bullet when it reaches the maximum height is

The y-component of vector $\vec{A}$ is $+3.0 \mathrm{~m}$ if $\vec{A}$ makes an angle of $30^{\circ}$ counter clockwise from the positive $y$ - axis, the magnitude of $\vec{A}$ is (assume $\vec{A}$ is in $x$ - y plane)

$\overrightarrow A ,\,\,\overrightarrow B ,\,\,\overrightarrow C \,$ are three coplanar vectors, then

In ground to ground projectile motion under gravity, which of the following doesn't affect the time of flight?

A bomb is released from a horizontal flying
aeroplane. The trajectory of bomb is

A person sitting in the rear end of the compartment throws a ball towards the front end. The ball follows a parabolic path. The train is moving with velocity of $20 \mathrm{~m} / \mathrm{s}$. A person standing outside on the ground also observes the ball. How will the maximum heights $\left({y}_{{m}}\right)$ attained and the ranges $({R})$ seen by the thrower and the outside observer compare with each other?

Three particles A, B and C are projected from the same point with the same initial speeds making angles ${30^0},{45^0}\,and\,{60^0}$ respectively with the horizontally. Which of the following statements is correct?

If a particle is projected with speed ${u}$ from the ground at an angle $\theta$ with horizontal. Radius of curvature of a point where velocity vector is perpendicular to initial velocity vector is given by

Consider three vectors $\mathbf{A}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}, \mathbf{B}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\mathbf{C}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$. A vector $\mathbf{X}$ of the form $\alpha \mathbf{A}+\beta \mathbf{B}$ ( $\alpha$ and $\beta$ are numbers) is perpendicular to $\mathbf{C}$. The ratio of $\alpha$ and $\beta$ is

Assertion : When the range of projectile is maximum, the time of flight is the largest.
Reason : Range is maximum when angle of projection is $45^\circ$

A ball rolls off the top of a stairway with horizontal velocity ${v_0}m{s^{ - 1}}$. If the steps are h metre high and w metre wide, the ball will hit the edge of nth step if

If $0.5\hat i + 0.8\hat j + \mathrm{c\hat k}$ is a unit vector, then the value of c is

Two particles are projected from the same point with the same speed, at different angles $\theta_1$ and $\theta_2$ to the horizontal. They have the same horizontal range. Their times of flight are $t_1$ and $t_2$ respectively incorrect statement is.

Vector $\vec{A}$ makes equal angles with $x, y$ and $z$-axis. Value of its components (in terms of magnitude of $\vec{A}$) will be