Motion in a Plane

Two vectors $\vec{A}$ and $\vec{B}$ are perpendicular. Their scalar product is:

What is the magnitude of vector $\vec{R} = 5\hat{i} + 12\hat{j}$?

The dot product is also known as:

Which law is used to find the resultant of two vectors originating from the same point?

If the dot product of two vectors is negative, it means the angle between them is:

Two non-zero vectors $\vec{A}$ and $\vec{B}$ are such that $|\vec{A} + \vec{B}| = |\vec{A}| + |\vec{B}|$. The angle between them is:

If $|\vec{A} \times \vec{B}|^2 + |\vec{A} \cdot \vec{B}|^2 = 144$ and $A = 4$, then $B$ is:

Two vectors $\vec{P}$ and $\vec{Q}$ are such that their resultant $\vec{R}_1$ is perpendicular to $\vec{Q}$. If the magnitude of vector $\vec{Q}$ is doubled while keeping its direction unchanged, the new resultant $\vec{R}_2$ is found to be perpendicular to the vector $\vec{P}$. What is the angle between the original vectors $\vec{P}$ and $\vec{Q}$?

Two vectors $\vec{a}$ and $\vec{b}$ are such that $|\vec{a} + \vec{b}| = |\vec{a}|$. Then the angle between $\vec{b}$ and $(2\vec{a} + \vec{b})$ is:

A force $\vec{F} = (5\hat{i} + 3\hat{j} + 2\hat{k})$ N is applied over a particle which displaces it from its origin to the point $\vec{r} = (2\hat{i} - \hat{j})$ m. The work done on the particle is:

Which of these is the correct unit vector notation for a vector $\vec{V}$ with magnitude $V$ and angle $\theta$ with the x-axis?

The component of a $10 \, N$ force along an axis inclined at $90^\circ$ to the force is:

If a vector $\vec{A}$ is resolved into $A_x$ and $A_y$, then the magnitude $A$ is given by:

A vector $\vec{D}$ has components $D_x = 3$ and $D_y = 3\sqrt{3}$. The angle made with the x-axis is:

The vector $\vec{P} = a\hat{i} + a\hat{j}$ makes an angle of _____ with the x-axis.

A force of $10\,N$ is resolved into two perpendicular components. If one component is $6\,N$, the other component is:

The component of a vector $\vec{A}$ along another vector $\vec{B}$ is zero. This implies that:

A vector is resolved into three mutually perpendicular components. If two of the components are $3$ and $4$ units and the magnitude of the vector is $13$ units, the third component is:

If a vector $\vec{P}$ of magnitude $10$ units is resolved into two components $A$ and $B$ such that the angle between $A$ and $B$ is $90^\circ$, what is the maximum value of $|A| + |B|$?

In a non-orthogonal coordinate system, a vector $\vec{A}$ is resolved into $A_1$ and $A_2$ along axes making angles $\theta_1$ and $\theta_2$ with $\vec{A}$. The magnitude $A$ is given by:

Ignoring air resistance, which component of the velocity vector remains constant throughout the flight?

A particle is projected with velocity $u$ at an angle $\theta$. The speed of the particle after time $t$ is given by:

The time of flight depends on which component of the initial velocity?

Which trigonometric function is involved in the calculation of the horizontal range ($R$)?

The speed of a projectile at its point of projection is $u$. What is its speed when it returns to the same horizontal level?

A projectile is thrown such that its range $R$ is $n$ times its maximum height $H$. The angle of projection is given by:

A body is projected with a speed $u$ at an angle $\theta$. Its speed when it is at a height equal to half of its maximum height is:

The condition for a projectile to have the same horizontal range for two different angles of projection $\theta_1$ and $\theta_2$ (with same $u$) is:

A projectile is thrown from the top of a tower of height $H$ with velocity $u$ horizontally. The distance from the foot of the tower where it strikes the ground is:

A particle is projected with velocity $u$ at an angle $\theta$ with the horizontal. The velocity at any time $t$ is $\vec{v}$. The rate of change of speed $dv/dt$ at $t = (u \sin \theta)/g$ is:

The ratio of angular speed of a second hand to a minute hand of a clock is:

In UCM, which of the following remains constant in magnitude but changes in direction?

If a particle moves in a circle of radius $R$ and its speed is $v$, the time taken for one revolution is:

The angular speed of the hour hand of a clock is:

Two cars are moving in concentric circles of radii $r_1$ and $r_2$ such that they complete one revolution in the same time. The ratio of their linear speeds $v_1 : v_2$ is:

A particle moves in a circle of radius $2\text{ m}$ with a constant speed. If it completes $5$ revolutions in $10\text{ s}$, its centripetal acceleration is:

A stone of mass $m$ is whirled in a horizontal circle of radius $r$ with speed $v$. If the tension in the string is $T$, then $T = mv^2/r$. If the speed is kept constant but the radius is doubled, the tension becomes:

Two strings of length $l$ are attached to a mass $m$. The other ends of the strings are attached to a vertical rod at a distance $d$ apart. When the rod rotates, the mass $m$ moves in a horizontal circle. If the tension in the upper string is $T_1$ and lower is $T_2$, and the mass is mid-way vertically between the attachment points, find $T_1 - T_2$.

A large sphere of radius $R$ is rotating with angular velocity $\omega$ about its vertical axis. At what latitude $\lambda$ will the effective acceleration due to gravity be zero, assuming $\omega^2 R > g$?

A flywheel rotates at a constant speed of $3000\text{ rpm}$. What is the angle described by a radius in $1\text{ second}$?

Vector multiplication by a real number $n$ affects:

If $n = 3$ and $m = 2$, then $n(m\vec{A})$ is equal to:

If $\vec{A} = \hat{i} + \hat{j}$, what is $-1(\vec{A})$?

If $n$ is positive, the angle between $\vec{A}$ and $n\vec{A}$ is:

Which of the following is the correct unit vector in the direction of vector $\vec{A}$?

Multiplying a vector $\vec{A}$ by -1 changes its:

If $\vec{A}$ and $\vec{B}$ are non-zero vectors and $\vec{A} = k\vec{B}$, then $\vec{A}$ and $\vec{B}$ are necessarily:

Consider $\vec{V} = \lambda ( \vec{A} + \vec{B} )$. If $\lambda = -1$, the vector $\vec{V}$ is:

If $\vec{A}$ is a vector and $k$ is a scalar, the magnitude of $k\vec{A}$ is always equal to:

In the equation $\vec{F} = m\vec{a}$, if the mass $m$ is considered a varying scalar (real number), and the acceleration $\vec{a}$ is constant, the force $\vec{F}$ will always:

In the graphical method, the subtraction $\vec{A} - \vec{B}$ can be visualized as adding $\vec{A}$ and a vector of the same magnitude as $\vec{B}$ but in the ________ direction.

Which of the following is NOT a property of vector addition?

Vectors that have opposite directions are called:

Vectors that have the same direction but different magnitudes are called:

In the following vector diagram, two different paths connect Point 1 to Point 3. Path 1 is formed by the head-to-tail arrangement of vectors $\vec{v_1}$ and $\vec{v_2}$, while Path 2 is formed by the head-to-tail arrangement of vectors $\vec{v_4}$ and $\vec{v_3}$. Based on the graphical method of vector addition, which relationship must hold true?

In the provided graphical representation, vectors $\vec{a}$ and $\vec{b}$ are drawn from a common origin $O$. If a vector $\vec{c}$ is drawn from the terminal point of $\vec{a}$ to the terminal point of $\vec{b}$, which equation correctly expresses $\vec{c}$ based on the triangle law of vector addition?

In the polygon law, if the polygon is not closed, the missing side that closes the polygon in the opposite sense represents:

Three vectors $\vec{A}, \vec{B}, \vec{C}$ satisfy $\vec{A} + \vec{B} = \vec{C}$ and $A^2 + B^2 = C^2$. Graphically, the angle between $\vec{A}$ and $\vec{C}$ is:

If the resultant of two forces of magnitudes $P$ and $Q$ is of magnitude $P$, what happens to the resultant if $P$ is doubled in magnitude (keeping the angle fixed)?

Given $|\vec{A}| = 3$, $|\vec{B}| = 4$. What are the maximum and minimum possible values for the magnitude of $\vec{A} - \vec{B}$?

When using the analytical method to add two vectors $\vec{A}$ and $\vec{B}$ of equal magnitude ($|\vec{A}| = |\vec{B}|$), if the angle between the two vectors is $\theta$, what is the angle $\alpha$ that the resultant vector $\vec{R}$ makes with vector $\vec{A}$?

The magnitude of the resultant of two vectors $\vec{A}$ and $\vec{B}$ is given by $\sqrt{A^2 + B^2}$ only when the angle between them is:

In the rectangular coordinate system, a vector $\vec{A}$ can be written as:

If the resultant of two vectors of magnitude $3$ and $4$ is $1$, the angle between them is:

The resultant of two forces has a magnitude of $20\,N$. One of the forces has a magnitude of $10\sqrt{3}\,N$ and makes an angle of $30^\circ$ with the resultant. Using the analytical method, calculate the magnitude of the second force.

Two vectors $\vec{A}$ and $\vec{B}$ have magnitudes of $3$ units and $6$ units respectively. If their resultant $\vec{R}$ is perpendicular to the vector $\vec{A}$, find the magnitude of the resultant $\vec{R}$ and the angle $\theta$ between the two vectors.

Two vectors $\vec{P}$ and $\vec{Q}$ act at an angle of $120^\circ$. The magnitude of $\vec{P}$ is $10$ units and that of $\vec{Q}$ is $15$ units. The magnitude of their resultant is approximately:

A vector $\vec{P}$ is added to a vector $\vec{Q}$. The resultant is $R_1$. If $\vec{Q}$ is replaced by its negative, the resultant is $R_2$. If $R_1 = R_2$, what is the angle between $\vec{P}$ and $\vec{Q}$?

If the angle between vectors $\vec{A}$ and $\vec{B}$ is $120^\circ$ and their magnitudes are 10 units each, the magnitude of $(\vec{A} - \vec{B})$ is:

Two vectors $\vec{A}$ and $\vec{B}$ are such that $|\vec{A} + \vec{B}| = |\vec{A}| + |\vec{B}|$. The angle between them is:

If a particle in a plane has $a_x = 2$ and $a_y = 0$, its velocity in the $y$-direction:

Which of the following is a vector quantity in 2D motion?

The magnitude of displacement in 2D motion is always:

In projectile motion, the angle between the velocity vector and the horizontal at the point of impact (on the same level) is:

A boat is sent across a river with a velocity of $8$ km/h. If the resultant velocity of the boat is $10$ km/h, then the velocity of the river is:

A projectile is thrown at an angle of $45^\circ$ with the horizontal. Its horizontal range is $B$. Its velocity at the highest point is:

The displacement of a particle is given by $\vec{r} = (t^2 - 4t + 6)\hat{i} + (t^2 - 6t + 5)\hat{j}$. The time at which the velocity vector is parallel to the $x$-axis is:

A particle moving with constant acceleration in a plane has a velocity $\vec{v}_1$ at time $t_1$ and $\vec{v}_2$ at time $t_2$. The acceleration vector is:

A particle moves in the $xy$-plane with a constant acceleration of $(3\hat{i} - 2\hat{j})\, \text{m/s}^2$. If the particle starts from the origin with an initial velocity of $(4\hat{i} + 6\hat{j})\, \text{m/s}$, at what time $t$ will its velocity be purely horizontal?

Two particles are projected simultaneously from the same point with velocities $u_1$ and $u_2$ at angles $\alpha_1$ and $\alpha_2$ with the horizontal. The path of one particle as seen from the other is a: