Motion in plane practice · Mixed difficulty

The time of flight of a projectile on an upward inclined plane depends upon

If unit vectors $\hat{a}$ and $\hat{b}$ are inclined at angle $\theta$, then $|\hat{a}-\hat{b}|=$

Which of the following is vector quantity?

The vectors $\mathbf{a}=2 \lambda^2 \hat{\mathbf{i}}+4 \lambda \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\mathbf{b}=7 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\lambda \hat{\mathbf{k}}$ make an obtuse angle whereas the angle between $\mathbf{b}$ and $\hat{\mathbf{k}}$ is acute and less than $\pi / 6$,

$\vec{A}=4 \hat{i}+3 \hat{j}$ and $\vec{B}=4 \hat{i}+2 \hat{j}$. Find a vector parallel to $\vec{A}$ but has magnitude five times that of $\vec{B}$.

Two particles are projected in air with same speed $u$ at angle $\theta_1$ and $\theta_2$ (both acute) to the vertical, respectively. If the maximum height reached by the first particle is equal to that of second, then which of the following is correct?

($T_1$ and $T_2$ are time of flight of two particles respectively)

A boat is moving with velocity $(4 \hat{i}+6 \hat{j}) \mathrm{m} / \mathrm{s}$ with respect to ground. The water in the river is moving with $(-3 \hat{i}-4 \hat{j}) \mathrm{m} / \mathrm{s}$ with respect to ground. The velocity of boat with respect to water is

If air resistance is not considered in projectiles, the horizontal motion takes place with

When a projectile is fired at an angle $\theta$ wrt. horizontal with velocity $u$, then its vertical component

Which of the following statements is/are incorrect?
(i) A scalar quantity is the one that is conserved in a process.
(ii) A scalar quantity is the one that can never take negative values.
(iii) A scalar quantity has the same value for observers with different orientations of the axes.

The vectors $\vec{A}$ and $\vec{B}$ are such that $|\overrightarrow{{A}}+\overrightarrow{{B}}|=|\overrightarrow{{A}}-\overrightarrow{{B}}|$. The angle between the two vectors is

A body projected with velocity $u$ at projection angle $\theta$ has horizontal range $R$. For the same velocity and projection angle, its range on the moon surface will be $\left(g_{\text {moon }}=g_{\text {earth }} / 6\right)$

At the highest point on the trajectory of a projectile, its

Assertion : The trajectory of projectile is quadratic in y and linear in x.
Reason : y component of trajectory is independent of x - component.

If $\vec{A}$ is perpendicular to $\vec{B}$, then choose the correct option:

If two vectors $\bar A = 2\hat i - 2\hat j + \hat k\,\,{\text{and }}\,\bar B = 2\hat i + 6\hat j - m\hat k$ are perpendicular to each other, then the value of $m$ is

Given below are two statements. One is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: Two identical balls A and B thrown with same velocity ’u’ at two different angles with horizontal attained the same range R. If A and B reached the maximum height ${h_1}$ and ${h_2}$ respectively, then $R = 4\sqrt {{h_1}{h_2}}$
Reason R: Product of said heights.
${h_1}{h_2} = \left( {\frac{{{u^2}{{\sin }^2}\theta }}{{2g}}} \right).\left( {\frac{{{u^2}{{\cos }^2}\theta }}{{2g}}} \right)$ Choose the correct answer:

In an oblique projectile motion if the velocity of projection is increased by 2%, the percent increase in horizontal range will be

During projectile motion, the horizontal
velocity

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

Magnitudes of four pairs of displacement vectors are given. Which pair of displacement vectors, under vector addition, fails to give a resultant vector of magnitude $3 {~ c m}$ ?

The two vectors $\overrightarrow{{A}}$ and $\overrightarrow{{B}}$ are drawn from a common point and $\overrightarrow{{C}}=\overrightarrow{{A}}+\overrightarrow{{B}}$, then angle between $\vec{A}$ and $\vec{B}$ is

(1) $90^{\circ}$ if $\mathrm{C}^2=\mathrm{A}^2+\mathrm{B}^2$

(2) greater than $90^{\circ}$ if $\mathrm{C}^2<\mathrm{A}^2+\mathrm{B}^2$

(3) greater than $90^{\circ}$ if $\mathrm{C}^2>\mathrm{A}^2+\mathrm{B}^2$

(4) less than $90^{\circ}$ if $\mathrm{C}^2>\mathrm{A}^2+\mathrm{B}^2$

Correct options are-

In the case of an oblique projectile, the velocity is perpendicular to acceleration

Assertion: To cross the river in minimum time, swimmer should swim in a direction perpendicular to the water current.
Reason: In this case river flow helps to cross the river.

If $\mathrm{\vec A}\mathrm{.\vec B} = 0\,and\,\,\mathrm{\vec A} \times \mathrm{\vec B} = 1$, then $\mathrm{\vec A}\;\,and\,\mathrm{\vec B}$ are

A body is projected from the earth at angle ${30^0}$ with the horizontal with some initial velocity. If its range is 20 m, the maximum height reached by its is (in metre )

If the unit vectors $\mathbf{e}_1$ and $\mathbf{e}_2$ are inclined at an angle $2 \theta$ and $\left|\mathbf{e}_1-\mathbf{e}_2\right|<1$, then for $\theta \in[0, \pi], \theta$ may lie in the interval

A body of mass $m$ is thrown at angle $\theta$ with horizontal from ground with initial velocity $v$. Find the angle of projection at a point where -

1. Velocity is half of initial velocity.

2. Velocity is $\frac{\sqrt{3}}{2}$ times of initial velocity

If $a, b, c$ and $A, B, C \in R-\{0\}$ such that $a A+b B+c C+\sqrt{\left(a^2+b^2+c^2\right)\left(A^2+B^2+C^2\right)}=0$, then value of $\frac{a B}{b A}=\frac{b C}{c B}+\frac{c A}{a C}$ is

Assertion: In a projectile motion, the acceleration is constant in both magnitude and direction but the velocity changes in both magnitude and direction.
Reason: When a force or acceleration is acting in an oblique direction to the direction of velocity, then both magnitude and direction of the velocity may be changed.