- \[\vec v_A\] = 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 [30]
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: 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: 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: 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: 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.
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.
Definition: Radial (Centripetal) Acceleration
The component of acceleration directed towards the centre of the circular path is called centripetal acceleration (or radial acceleration).
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: 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: Angular Acceleration (α)
The rate of change of angular velocity of a body is called angular acceleration.
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: 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: 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: Modulus of a Vector
The length or the magnitude of a vector is called the modulus of a vector.
Definition: Unit Vector
A vector of unit magnitude drawn in the direction of a given vector is called a unit vector.
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: Co-planar Vectors
The vectors which act in the same plane are called co-planar vectors.
Definition: Instantaneous Velocity
Instantaneous velocity of an object is its velocity at a given instant of time. It is defined as the limiting value of the average velocity of the object over a small time interval (Δt) around t when the value of the time interval (Δt) goes to zero.
OR
The limiting value of the average velocity of an object over a small time interval 'Δt' around time t when the value of the time interval goes to zero is called instantaneous velocity.
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: Acceleration
Acceleration is defined as the rate of change of velocity with time.
OR
The rate of change of velocity with respect to time — a vector quantity whose direction is the same as that of change in velocity, with dimensional formula [M0L1T−2] and SI unit m/s² — is called acceleration.
Definition: Average Acceleration
The change in velocity of an object divided by the total time required for that change in velocity is called average acceleration.
OR
The ratio of total change in velocity to the total time taken by the particle when the change in velocity results is called average acceleration.
Definition: Instantaneous Acceleration
The limiting value of the average acceleration of an object over a small time interval 'Δt' around time tt when the value of the time interval goes to zero is called instantaneous acceleration.
OR
The acceleration of a particle at a particular instant of time — defined as the limit of average acceleration as time interval Δt→0 — is called instantaneous acceleration.
Definition: Uniform Acceleration
The acceleration when the magnitude and direction of the acceleration remains constant during motion of an object is called uniform acceleration.
Definition: Non-Uniform Acceleration
The acceleration when either magnitude or direction or both change during motion is called non-uniform acceleration.
Definition: Gravitational Acceleration
The acceleration on an object which results due to gravity — where every small body accelerates in a gravitational field at a similar rate towards the centre of mass, irrespective of the mass of the body — is called gravitational acceleration.
Definition: Retardation / Deceleration
The negative acceleration (i.e., uniformly retarded motion where a < 0) that shows slowing down or deceleration of a particle is called retardation.
Formulae [9]
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: 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: Instantaneous velocity
\[\vec{\mathrm{v}}=\lim_{\Delta t\to0}\left(\frac{\Delta\vec{x}}{\Delta t}\right)=\frac{d\vec{x}}{dt}\]
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: Average Acceleration
Average acceleration is calculated when an object has velocities \[\vec v_1\] and \[\vec v_2\] at times t1 and t2:
\[\vec{a}=\frac{\vec{v_2}-\vec{v_1}}{t_2-t_1}\]
where:
- \[\vec a\] = average acceleration
- \[\vec v_1\] = velocity at time t1
- \[\vec v_2\] = velocity at time t2
OR
Average acceleration: \[\vec a_{av}=\frac {\vec v_2-\vec v_1}{t_{2}-t_{1}}=\frac {\Delta\vec v}{\Delta t}\]
Formula: Instantaneous Acceleration
Instantaneous acceleration is the limiting value of average acceleration when the time interval approaches zero:
\[\vec{a}=\lim_{\Delta t\to0}\frac{\Delta\vec{v}}{\Delta t}=\frac{d\vec{v}}{dt}\]
where:
- \[\vec a\] = instantaneous acceleration
- \[d\vec{v}\] = infinitesimal change in velocity
- dt = infinitesimal change in time
The instantaneous acceleration at a given time equals the slope of the tangent to the velocity versus time curve at that time.
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: Vector Analysis
- Distance vs Displacement: Distance (5 km) is scalar; displacement (5 km north) is vector.
- Speed vs Velocity: Speed (60 km/h) is scalar; velocity (60 km/h north) is vector.
- Vectors add differently: You cannot simply add vectors like scalars. A 5 N force east + 5 N force north ≠ 10 N!
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 = \[\sqrt{A^2+B^2+2AB\cos\theta}\].
-
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: 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 [18]
- Vector Analysis
- 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
- Uniform Circular Motion (UCM)
- Vector
- Instantaneous Velocity
- Rectangular Components
- Scalar (Dot) and Vector (Cross) Product of Vectors
- Relative Velocity in Two Dimensions
- Cases of Uniform Velocity
- Cases of Uniform Acceleration Projectile Motion
- Acceleration in Linear Motion
- Angular Velocity
- Introduction of Motion in One Dimension
