Motion in plane practice · Mixed difficulty

A particle moves in $X Y$-plane from positions $(2 \mathrm{~m}, 4 \mathrm{~m})$ to $(6 \mathrm{~m}, 8 \mathrm{~m})$ is $2 \mathrm{~s}$. Magnitude and direction of average velocity is

Can scalar product be ever negative?

Let A, B and $\mathbf{C}$ be unit vectors. Suppose $\mathbf{A} \cdot \mathbf{B}=\mathbf{A} \cdot \mathbf{C}=0$ and the angle between $\mathbf{B}$ and $\mathbf{C}$ is $\frac{\pi}{4}$. Then,

Which of the following statement is correct?

Assertion: A coin is allowed to fall in a train moving with constant velocity. Its trajectory is a straight line as seen by an observer attached to the train.
Reason: An observer on ground will see the path of coin as a parabola.

Two vectors are perpendicular, if

Assertion: Generally the path of a projectile from the earth is parabolic but it is elliptical for projectiles going to a very great height.

Reason: Up to ordinary height the projectile moves under a uniform gravitational force, but for great heights, projectile moves under a variable force.

The horizontal and vertical displacements $x$ and $y$ of projectile at a given time $t$ are given by $x = 12\,t\,m$ and $y = 5t - 5{t^2}\,m$. The range of the projectile in metres is

Which of the following is not true? If $\vec{A}=3 \hat{i}+4 \hat{j}$ and $\vec{B}=6 \hat{i}+8 \hat{j}$ where $A$ and $B$ are the magnitudes of $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{B}}$

The resultant of two forces 3P and 2P is R. If the first force is doubled them the resultant is also doubled. The angle between the two forces is

A force is inclined at $50^{\circ}$ to the horizontal. If its rectangular component in the horizontal direction be $50 \mathrm{~N}$, find the magnitude of the force and its vertical component.

Assertion: Adding a scalar to a vector of the same dimensions is a meaningful algebraic operation.
Reason : Displacement can be added to distance.

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.

Assertion A: When a body is projected at an angle $45^{\circ},$ it's range is maximum.

Reason R: For maximum range, the value of $\sin 2 \theta$ should be equal to one.

In the light of the above statements, choose the correct answer from the options given below:

The resultant of the tow vectors having magnitude 2 and 3 is 1. What is their cross product.

It is found that $\left| {A + B} \right| = \left| A \right|,$ this necessarily implies

One fielder throws a ball and in 4 secs it reaches to another player. So, the maximum height reached by the ball above the point of projection would be about

A projectile is fired at an angle of 30° to the horizontal such that the vertical component of its initial velocity is $80\mathrm{\;m}{\mathrm{s}^{ - 1}}$. Its time of flight is T. Its velocity at $t = \frac{T}{4}$ has a magnitude of nearly

The angle between vector $\overrightarrow{Q}$ and the resultant of $(2 \overrightarrow{Q}+2 \overrightarrow{{P}})$ and $(2 \overrightarrow{{Q}}-2 \overrightarrow{{P}})$ is:

Two forces of $5 \mathrm{~N}$ and $7 \mathrm{~N}$ act on a particle with an angle of $60^\circ$ between them. Find the resultant force.

The torque force $\bar F = 2i - 3j + 4k$ newtons acting at the point $\bar r = 3i + 2j + 3k$ meters about the origin is

Choose the wrong statement.

An ant starts from the origin and crawls $10 \mathrm{~cm}$ along the $x$ - axis and then $20 \mathrm{~cm}$ along the $y-$ axis. The dot product of the ant's displacement vector with the position vector of a point that makes $45^{\circ}$ with the $x$ - axis and has a magnitude of $\sqrt{2} \mathrm{~cm}$ is

$(\mathbf{a} \cdot \hat{\mathbf{i}}) \hat{\mathbf{i}}+(\mathbf{a} \cdot \hat{\mathbf{j}}) \hat{\mathbf{j}}+(\mathbf{a} \cdot \hat{\mathbf{k}}) \hat{\mathbf{k}}$ is equal to

Minimum number of vectors of unequal magnitude whose vector sum can equal to zero is

Given that $\vec{C}=\vec{A}+\vec{B}$ and $\vec{C}$ makes an angle $\alpha$ with $\vec{A}$ and $\beta$ with $\vec{B}$. Which of the following options is correct?

A body is projected at $30^{\circ}$ with the horizontal. The air offers resistance in proportion to the velocity of the body. Which of the following statements is correct?

What is the angle between $(\mathbf{P}+\mathbf{Q})$ and $(\mathbf{P} \times \mathbf{Q})$ ?

Select the statements which are not true regarding the vectors?

For motion in two or three dimensions, the angle between velocity and acceleration is:

A vector $\mathbf{a}$ is turned without a change in its length through a small angles $d \theta$. The value of $|\Delta \mathbf{a}|$ and $\Delta a$ are, respectively.