Motion in plane practice · Mixed difficulty

Assertion: Flight path of an object moving under the acceleration due to gravity can be straight line or a parabola depending on the initial conditions.
Reason: The shape of the flight path of the motion of an object is determined by the acceleration only.

When forces $\vec{P}_1, \vec{P}_2, \vec{P}_3$ are acting on a particle of mass $(m)$ such that $\vec{P}_2$ and $\vec{P}_3$ are mutually perpendicular if particle remains stationary then

If none of the vectors $\vec{A}, \vec{B}$ and $\vec{C}$ are zero and if $\vec{A} \times \vec{B}=0$ and $\overrightarrow{\mathrm{B}} \times \overrightarrow{\mathrm{C}}=0$, then the value of $|\vec{A} \times \vec{C}|$ is

The projection of $\mathbf{a}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ on $\mathbf{b}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ is

The equation of projectile is $y = 12x - \frac{{{x^2}}}{6}$. The horizontal range is

A hiker stands on the edge of a cliff $490 \mathrm{~m}$ above the ground and throws a stone horizontally with an intial speed of $15 \mathrm{~m} / \mathrm{s}$. Neglecting air resistance. Find the time taken by the stone to reach the ground, and the speed with which it hits the ground $\left({g}=9.8 \mathrm{~m} / \mathrm{s}^2\right)$:

Statement I: For angle of projection $\tan ^{-1}(4)$, the horizontal range and maximum height are equal.

Statement II: The maximum range of projectile is directly proportional to square of velocity and inversely proportional to acceleration of gravity.

Assertion: If the initial and final positions coincide, the displacement is a null vector.
Reason: A physical quantity cannot be called a vector, if its magnitude is zero.

If $x, y$ and $z$ are distinct scalars such that $[x \mathbf{a}+y \mathbf{b}+z \mathbf{c}, x \mathbf{b}+y \mathbf{c}+z \mathbf{a}, x \mathbf{c}+y \mathbf{a}+z \mathbf{b}]=0]$ where $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are non-coplanar vectors, then

A particle has been projected with a speed of $20 \mathrm{~ms}^{-1}$ at an angle of $30^{\circ}$ with the horizontal. The time taken when the velocity vector becomes perpendicular to the initial velocity vector is

The sum and difference of two perpendicular vectors of equal length are

The horizontal and vertical displacements of a projectile, as a function of time, are given by $x = 10\,t\,m$ and $y = 12t - 4{t^2}\,m$. The horizontal range of the projectile is

Two particles are projected with same velocity but at angles of projection $25^\circ \,$ and $65^\circ \,$ with horizontal the ratio of their horizontal ranges is

A grasshopper can jump a maximum distance of $1.6 \mathrm{~m}$. How far can it go in $10 \mathrm{~s}$ ?

A vector multiplied by the number 0, results into

Train $A$ is moving along two parallel rail tracks towards north with speed $72 \mathrm{~km} / \mathrm{h}$ and train $B$ is moving towards south with speed $108 \mathrm{~km} / \mathrm{h}$. Velocity of train $B$ with respect to $A$ and velocity of ground with respect to $B$ are (in $\mathrm{ms}^{-1}$)

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

Identify the vector quantity among the following

Assertion (A) : In projectile motion if particle is projected with speed $u$, then speed of particle at height $h$ would be $\sqrt{u^2-2 g h}$.

Reason (R) : If particle is projected with vertical component of velocity $u_y$. Then vertical component at the height $h$ would be $\pm \sqrt{u_y^2-2 g h}$

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: $|\vec{a}+\vec{b}|^2=a^2+b^2+2\vec{a} \cdot \vec{b}$

If $\vec{a}=(\hat{i}+2 \hat{j}+2 \hat{k})$ and $\vec{b}=(2 \hat{i}+\hat{j}+2 \hat{k})$. The projection of $\vec{b}$ on $\vec{a}$ is

If $\mathrm{\vec A} = 2\hat i + 3\hat j + 6\hat k\,and\,\mathrm{\vec B} = 3\hat i - 6\hat j + 2\hat k$ then vector perpendicular to both $\mathrm{\vec A}\,and\,\mathrm{\vec B}$ has magnitude k times that $\left( {6\hat i + 2\hat j - 3\hat k} \right)$. That k is equal to

Given that A and B are greater than 1. The magnitude of $\left( {\mathrm{\vec A} \times \mathrm{\vec B}} \right)$ can not be

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

A projectile is thrown with initial velocity $\left( {{V_X}\hat i + {V_Y}\hat j} \right)\,m/s$. If the range of projectile is double the maximum height reached then $\left( {\frac{{{V_y}}}{{{V_x}}}} \right)$ is

If $x, y$ and $z$ are distinct scalars such that $[x \mathbf{a}+y \mathbf{b}+z \mathbf{c}, x \mathbf{b}+y \mathbf{c}+z \mathbf{a}, x \mathbf{c}+y \mathbf{a}+z \mathbf{b}]=0]$ where $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are non-coplanar vectors, then

A bullet is dropped from the same height when another bullet is fired horizontally. They will hit the ground

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.

The components of a vector along the $x$ and $y$ directions are $\left( {n + 1} \right)$ and 1 respectively. If the coordinate system is rotated by an angle ${60^0}$, then the components changes to $n$ and 3. Find the value of $n$.

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