- \[\vec v_A\] = Velocity of object A
- \[\vec v_B\] = Velocity of object B
- \[\vec v_{AB}\] = Velocity of A relative to B
- The subscripts indicate the order: for \[\vec v_{AB}\], A is the observer, B is referenced.
Definitions [39]
Definition: Motion in a Plane
The motion of an object in which its position changes along two mutually perpendicular axes (X and Y) simultaneously, such that it requires two coordinates to describe its position at any given instant, is called Motion in a Plane (Two-Dimensional Motion).
OR
The motion of a body that takes place along two different directions (or co-ordinate axes) with respect to origin at the same time is called two dimensional motion or motion in a plane.
Definition: Scalar Quantity
The physical quantities which have only magnitude and no direction, and can be specified by a single number along with a proper unit (e.g., mass, volume, density, time, temperature, electric current) are called scalar quantities or scalars.
Definition: Position Vector
A vector which gives the position of an object with reference to the origin of a coordinate system is called a position vector.
Definition: Displacement Vector
The vector which tells how much and in which direction an object has changed its position in a given time interval is called a displacement vector.
Definition: Equal Vectors
Two vectors that have the same magnitude and the same direction are called equal vectors.
Definition: Scalar (Dot) Product of Two Vectors
The scalar or dot product of two vectors \[\vec A\] and \[\vec B\] defined as the product of the magnitudes of \[\vec A\] and \[\vec B\] and cosine of the angle θ between them is called the scalar product or dot product.
Definition: Vector (Cross) Product of Two Vectors
The vector or cross product of two vectors is defined as the vector whose magnitude is equal to the product of the magnitudes of two vectors and sine of the angle between them, and whose direction is perpendicular to the plane of the two vectors and is given by the right hand rule — this is called the vector product or cross product.
Definition: Orthogonal Triad of Base Vectors
The three mutually perpendicular unit vectors \[\hat i\], \[\hat j\], \[\hat k\] used in three-dimensional space to describe the direction of any vector — where \[\hat i\] is along X-axis, \[\hat j\] along Y-axis, and \[\hat k\] along Z-axis — are called an orthogonal triad of base vectors.
Definition: Direction Cosines
The values of cosα, cosβ, and cosγ which are the cosines of the angles subtended by the rectangular components with the given vector are called direction cosines of a vector.
Definition: Component Vectors
The splitting vectors obtained when a single vector is resolved into two or more vectors in different directions are called component vectors.
Definition: Resolution of the Vector
A vector \[\vec V\] can be expressed as the sum of two or more vectors along fixed directions. This process is known as vector resolution.
OR
The process of splitting a single vector into two or more vectors in different directions which together produce same effect as produced by the single vector alone is called resolution of vector.
Definition: Rectangular Components
When a vector is resolved into components along mutually perpendicular directions (like x and y axes in 2D, or x, y, and z axes in 3D), these components are called rectangular or Cartesian components.
OR
When a vector is resolved along two mutually perpendicular directions, the components so obtained are called rectangular components of the given vector.
Definition: Maximum Height
The maximum vertical height reached by the projectile — i.e., the distance travelled along the vertical (y) direction up to the highest point — is called the maximum height (H).
Definition: Projectile
An object in flight after being thrown with some velocity that follows a curved path under the action of gravity is called a projectile.
OR
A body in free fall which is subjected to the force of gravity and air resistance only — which refers to the motion of bodies flung into the air — is called a projectile.
Definition: Time of Flight
The total time for which the projectile remains in the air — from the moment it is projected to the moment it returns to the same level — is called the time of flight (T).
Definition: Time of Descent
The time taken by the projectile to travel from the maximum height back to the ground is called the time of descent (tD).
Definition: Trajectory
The path followed by a projectile is called its trajectory.
Definition: Projection Angle
The direction of projection with respect to the horizon which determines the shape of trajectory (vertical → vertical, oblique → parabolic, horizontal → half parabolic) is called the projection angle.
Definition: Projection Speed
The magnitude of projection velocity — which, with a fixed projection angle, shows the length of trajectory or range — is called the projection speed.
Definition: Centripetal Acceleration
The acceleration acting on an object undergoing uniform circular motion, which always acts on the object along the radius towards the centre of the circular path, is called centripetal acceleration.
Definition: Time of Ascent
The time taken by the projectile to travel from the point of projection to the maximum height is called the time of ascent (tA).
Definition: Horizontal Range
The total maximum horizontal distance travelled by a projectile from the point of projection to the point where it hits the ground is called the horizontal range (R).
Define Uniform circular motion.
When a particle moves with a constant speed in a circular path, its motion is said to be the uniform circular motion.
Define angular velocity.
Angular velocity of a particle is the rate of change of angular displacement.
Definition: Uniform Circular Motion
When a particle moves with a constant speed in a circular path, its motion is said to be uniform circular motion.
OR
The motion of a body moving with constant speed along a circular path is called uniform circular motion.
OR
The motion of a body moving with constant speed along a circular path, where the velocity is always tangential to the circular path and remains constant in magnitude, is called uniform circular motion.
Definition: Angular Displacement
The angle traced out by the radius vector at the centre of the circular path in a given time, expressed as Δθ = θ2 − θ1, is called angular displacement.
Definition: Angular Velocity (ω)
The rate of change of angular displacement of a body undergoing circular motion is called angular velocity.
Definition: Angular Acceleration (α)
The rate of change of angular velocity of a body is called angular acceleration.
Definition: Radial (Centripetal) Acceleration
The component of acceleration directed towards the centre of the circular path is called centripetal acceleration (or radial acceleration).
Definition: Centripetal Force
The force directed towards the centre along the radius, required to keep a body moving along a circular path at constant speed, is called centripetal force.
Definition: Modulus of a Vector
The length or the magnitude of a vector is called the modulus of a vector.
Definition: Negative of a Vector
A vector having the same magnitude as the original vector but having an opposite direction is called the negative of a vector.
Definition: Co-planar Vectors
The vectors which act in the same plane are called co-planar vectors.
Definition: Zero (Null) Vector
A vector that has zero magnitude and an arbitrary direction, represented by \[\vec 0\], is called a zero vector or null vector.
Definition: Unit Vector
A vector of unit magnitude drawn in the direction of a given vector is called a unit vector.
Definition: Vector
A vector is any quantity that needs both magnitude (size) and direction to be completely described.
OR
The physical quantities which have both magnitude and direction, obey the laws of vector addition, and are specified by a number with a unit and its direction (e.g., displacement, velocity, force, momentum) are called vector quantities or vectors.
Definition: Relative Velocity
The velocity of an object with respect to another object — i.e., the velocity with which an object appears to move to an observer which is placed on the other object that moves along with it — is called relative velocity.
Definition: Scalar Product
The scalar product or dot product of two nonzero vectors \[\vec P\] and \[\vec Q\] is defined as the product of the magnitudes of the two vectors and the cosine of the angle θ between the two vectors.
Definition: Vector Product
The Vector Product (or Cross Product) is a method of multiplying two vectors (\[\vec P\] and \[\vec Q\]) that results in a new vector (\[\vec R\]). This new vector is fundamentally related to the rotation or perpendicular effects created by the two original vectors.
The magnitude of the resulting vector R is defined by the product of the magnitudes of the two vectors and the sine of the smaller angle (θ) between them.
Magnitude: ∣R∣ = ∣ P × Q ∣ = PQ sin θ
Formulae [14]
Formula: Position Vector in x-y Plane
\[\vec{r}=x\hat{i}+y\hat{j}\]
Formula: Displacement
\[\Delta\vec{r}=\vec{r^{\prime}}-\vec{r}=(x^{\prime}-x)\hat{i}+(y^{\prime}-y)\hat{j}=\Delta x\hat{i}+\Delta y\hat{j}\]
Formula: Average Velocity
\[\bar{v}=\frac{\Delta\vec{r}}{\Delta t}\]
Formula: Instantaneous Velocity
\[\bar{v}_{inst}=\lim_{\Delta t\to0}\frac{\Delta\vec{r}}{\Delta t}=\frac{d\vec{r}}{dt}\]
\[\vec{v}=v_x\hat{i}+v_y\hat{j}+v_z\hat{k}\],
where vx = \[\frac {dx}{dt}\], vy = \[\frac {dy}{dt}\], vz = \[\frac {dz}{dt}\]
Fomula: Instantaneous Acceleration
\[a_{inst}=\lim_{\Delta t\to0}\frac{\Delta\vec{v}}{\Delta t}=\frac{d\vec{v}}{dt}\]
\[\vec{a}=a_x\hat{i}+a_y\hat{j}+a_z\hat{k}\],
where ax = \[\frac {dv_x}{dt}\], ay = \[\frac {dv_y}{dt}\], az = \[\frac {dv_z}{dt}\]
Formula: Magnitude of a 3D Vector
The magnitude of vector \[\vec A\] resolved into three-dimensional components is:
A = \[\sqrt{A_x^2+A_y^2+A_z^2}\]
Formula: Identity of Direction Cosines
The sum of squares of all direction cosines is always equal to 1:
cos2α + cos2β + cos2γ = 1
Formula: Direction Cosines
If α, β, and γ are the angles subtended by the rectangular components with the given vector, then:
cos α = \[\frac {A_x}{A}\], cos β = \[\frac {A_y}{A}\], cos γ = \[\frac {A_z}{A}\]
Formula: Resolution of a Vector
When a vector \[\vec A\] is resolved into three-dimensional rectangular components, it is given by:
\[\vec A\] = Ax\[\hat i\] + Ay\[\hat j\] + Az\[\hat k\]
For resultant of multiple vectors resolved along axes:
X = ∑Fi cosθi, Y = ∑Fi sin θi
F = \[\sqrt {X^2+Y^2}\], ϕ = tan−1(\[\frac {Y}{X}\])
Formula: Projectile Motion
| Quantity | Formula |
|---|---|
| Position after time t | x = (u cos θ)t, y = (u sin θ)t − \[\frac {1}{2}\]gt2 |
| Equation of trajectory | y = x tan θ − \[\frac {g}{2u^2 cos^2 θ}\] ⋅ x2 |
| Maximum height | H = \[\frac {u^2 sin^2 θ}{2g}\] |
| Time of flight | T = \[\frac {2u sin θ}{g}\] |
| Horizontal range | R = \[\frac {u^2 sin 2θ}{g}\] |
| Maximum range | Rmax = \[\frac {u^2}{g}\] at θ = 45° |
| Velocity after time ttt | vx = u cos θ, vy = u sin θ − gt |
| Speed | v = \[\sqrt {v_x^2+v_y^2}\] |
Formula: Velocity of A relative to B
\[\vec{v}_{AB}=\vec{v}_A-\vec{v}_B\]
where:
Formula: Velocity of B relative to A
\[\vec{v}_{BA}=\vec{v}_B-\vec{v}_A\]
where:
Formula: Scalar (Dot) Product
\[\vec{A}\cdot\vec{B}=|\vec{A}||\vec{B}|\cos\theta=AB\cos\theta\]
Formula: Vector (Cross) Product
\[\vec{A}\times\vec{B}=AB\sin\theta\hat{n}\]
Theorems and Laws [4]
Law: Right Hand Thumb Rule
The rule which states that if we curl the fingers of the right hand in such a way that they point in the direction of rotation from vector \[\vec A\] to \[\vec B\] through the smaller angle, then the stretched thumb points in the direction of \[\vec A\] × \[\vec B\] is called the Right Hand Thumb Rule.
Law: Triangle Law of Vector Addition
If two vectors can be represented both in magnitude and direction by the two sides of a triangle taken in the same order, then their resultant is represented both in magnitude and direction by the third side of the triangle taken in the opposite order — this is called the Triangle Law of Vector Addition.
Law: Parallelogram Law of Vector Addition
If two vectors can be represented both in magnitude and direction by the two adjacent sides of a parallelogram drawn from a common point, then their resultant is completely represented, both in magnitude and direction, by the diagonal of the parallelogram passing through that point — this is called the Parallelogram Law of Vector Addition.
Law: Polygon Law of Vector Addition
If a number of vectors are represented both in magnitude and direction by the sides of an open polygon taken in the same order, then their resultant is represented both in magnitude and direction by the closing side of the polygon taken in opposite order — this is called the Polygon Law of Vector Addition.
Key Points
Key Points: Addition and Subtraction of Vectors
-
Component Method: Resultant R = A + B is found as Rx = Ax + Bx, Ry = Ay + By, Rz = Az + Bz, giving R = Rx\[\hat i\] + Ry\[\hat j\] + Rz\[\hat k\].
-
Laws of Addition: Triangle law (head-to-tail), Parallelogram law (tail-to-tail, diagonal = resultant), and Polygon law (for multiple vectors, closing side = resultant).
-
Magnitude (Addition): When A and B are at angle θ, R = \[\sqrt{A^2+B^2+2AB\cos\theta}\].
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Magnitude (Subtraction): Change the sign to minus — ∣R∣ = .
-
Direction of Resultant: tanα = \[\frac{B\sin\theta}{A+B\cos\theta}\] for addition; tanβ = \[\frac{B\sin\theta}{A-B\cos\theta}\] for subtraction.
Key Points: Projectile Motion
- Horizontal range is maximum at 45° and reduces for any other angle of projection.
- A projectile has two simultaneous independent motions — constant horizontal and gravity-driven vertical.
- The path is a symmetric parabola — equal time up and down, equal speed at the same height.
Key Points: Uniform Circular Motion
- In UCM, speed is constant, but velocity continuously changes direction, always remaining tangential to the path.
- Angular displacement is the angle swept by the radius vector; angular velocity is its rate of change.
- Even at constant speed, centripetal acceleration is never zero — it always acts towards the centre of the circular path.
- Centripetal force is always directed towards the centre and is essential to maintain circular motion — it does no work on the body.
- If speed is constant in circular motion, tangential acceleration = 0, but radial acceleration ≠ 0.
Concepts [16]
- Motion in Two Dimensions - Motion in a Plane
- Scalars and Vectors
- Position and Displacement Vectors
- Equality of Vectors
- Multiplication of Vectors by a Real Number or Scalar
- Vector Operations>Addition and Subtraction of Vectors
- Resolution of Vectors
- Vector Addition – Analytical Method
- Motion in a Plane
- Equations of Motion in a Plane with Constant Acceleration
- Projectile Motion
- Uniform Circular Motion (UCM)
- Vector
- Relative Velocity in Two Dimensions
- Multiplication of Vectors>Scalar Product(Dot Product)
- Multiplication of Vectors>Vector Product (Cross Product)
