- \[\vec v_A\] = 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 [26]
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.
1. Uniform motion
“In physics, uniform motion is defined as the motion where the velocity of the body travelling in a straight line remains the same. When the distance travelled by a moving thing is the same at several time intervals, regardless of the time length, the motion is said to be uniform motion.”
For example,
- The hour hand of the clock: It moves with uniform speed, completing movement of a specific distance in an hour.
- An aeroplane is cruising at a level height and a steady speed.
- A car is going along a straight, level road at a steady speed.
2. Non-Uniform motion
Non-uniform motion is used to mean the movement in which the object does not cover the same distance in the same distances in the same time intervals, regardless of the length of the time intervals. Every time the speed of the moving object changes by a different proportion at the same time interval, the motion of the body is observed as non-uniform motion.
For example:
- A horse running.
- A bouncy ball.
- A car coming to a halt.
Definition: Average Velocity
"Average velocity is defined as the displacement of the object during the time interval over which average velocity is being calculated, divided by that time interval."
OR
The total displacement Δ\[\vec x\] of an object divided by the total time interval Δt over which that displacement occurs is called average velocity.
OR
The ratio of total displacement to the total time taken by the body is called average velocity.
Definition: Co-planar Vectors
The vectors which act in the same plane are called co-planar vectors.
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: 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: 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: 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: 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: 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: 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: 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: 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: 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.
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.
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: 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: Radial (Centripetal) Acceleration
The component of acceleration directed towards the centre of the circular path is called centripetal acceleration (or radial acceleration).
Formulae [9]
Formula: Instantaneous velocity
\[\vec{\mathrm{v}}=\lim_{\Delta t\to0}\left(\frac{\Delta\vec{x}}{\Delta t}\right)=\frac{d\vec{x}}{dt}\]
Formula: Average Velocity
\[\vec{v}_{\mathrm{av}}=\frac{\vec{x}_2-\vec{x}_1}{t_2-t_1}\]
- vav : average velocity.
- x2 : final position vector.
- x1 : initial position vector.
- t2 : final time
- t1 : initial time
Dimensions: [L1M0T−1]
OR
Average Velocity: \[\vec V_{avg}\] = \[\frac {\text {Displacement}}{\text {Time interval}}\] = \[\frac {x_2-x_1}{t_2-t_1}\] = \[\frac {Δ\vec x}{Δt}\]
Formula: Velocity
Velocity = \[\frac {\text {Displacement}}{\text {Time interval}}\]
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: 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: 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: 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: Identity of Direction Cosines
The sum of squares of all direction cosines is always equal to 1:
cos2α + cos2β + cos2γ = 1
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: Position-time, Velocity-time and Acceleration-time Graphs
- Slope of x-t graph = Velocity → Horizontal line (x = constant) = rest; positive slope = uniform velocity along +X axis; negative slope = motion along −X axis; curve = non-uniform velocity.
- Slope of v-t graph = Acceleration → Horizontal line = zero acceleration (constant velocity); positive slope = constant positive acceleration; negative slope = constant negative acceleration; curve = non-uniform acceleration.
- Area under v-t graph = Displacement of the object during that time interval.
- x-t graph shapes → Zigzag/oscillatory curve = oscillatory motion with constant speed; ever-increasing curve = accelerated (non-uniform) motion.
- Relative Velocity → \[\vec v_{AB}\] = \[\vec v_A\] − \[\vec v_B\] and \[\vec v_{BA}\] = \[\vec v_B\] − \[\vec v_A\]; they are always equal in magnitude and opposite in direction.
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 [20]
- Position, Path Length and Displacement
- Position - Time Graph
- Instantaneous Velocity
- Elementary Concept of Differentiation and Integration for Describing Motion
- Uniform and Non-uniform Motion
- Average Velocity
- Uniformly Accelerated Motion
- Position-time, Velocity-time and Acceleration-time Graphs
- Relations for Uniformly Accelerated Motion (Graphical Treatment)
- Differentiation as Rate of Change
- Vector Analysis
- Vector
- Multiplication of Vectors by a Real Number or Scalar
- Vector Operations>Addition and Subtraction of Vectors
- Relative Velocity in Two Dimensions
- Resolution of Vectors
- Rectangular Components
- Scalar (Dot) and Vector (Cross) Product of Vectors
- Motion in a Plane
- Uniform Circular Motion (UCM)
