- \[\vec v_A\] = 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 [36]
Defination: Speed
Speed is the distance travelled by an object in a given amount of time without considering the direction.
Formula: Speed = `"Distance traversed" / "Total time."`
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.
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.
Definition: Average Speed
"total path length travelled during the time interval over which average speed is being calculated, divided by that time interval."
OR
The total distance travelled by an object divided by the total time taken for its motion is called average speed.
OR
The ratio of total distance travelled by the body to the total time taken to cover such distance is called average speed.
Definition: Non-Uniform Speed (Variable Speed)
The speed at which an object covers unequal distances in equal intervals of time is called non-uniform speed or variable speed.
Definition: Uniform Speed
The speed at which an object covers equal distances in equal intervals of time is called uniform speed.
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: 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: 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: 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: 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: 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: 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: 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: 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).
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: 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: 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: 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: 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.
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: 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.
Formulae [9]
Formula: Average Speed
Average Speed = vav = \[\frac{\text{path length}}{\text{time interval}}\]
OR
Average speed = \[\frac {\text {Total path length}}{\text {Total time int erval}}\] = \[\frac {\text {Total distance}}{\text {Total time}}\] = \[\frac {x}{t}\]
Formula: Speed
Speed = \[\frac {Distance covered}{t}\] = \[\frac {s}{t}\]
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: 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
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:
Theorems and Laws [3]
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 = \[\sqrt{A^2+B^2+2AB\cos\theta}\].
-
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 [20]
- Introduction to Kinematics
- Position, Path Length and Displacement
- Position - Time Graph
- Speed and Velocity
- Uniform and Non-uniform Motion
- Average Speed
- Uniformly Accelerated Motion
- Velocity - Time Graphs
- Relations for Uniformly Accelerated Motion (Graphical Treatment)
- Vector Analysis
- Vector
- Vector Operations>Addition and Subtraction of Vectors
- Vector Addition – Analytical Method
- Scalar (Dot) and Vector (Cross) Product of Vectors
- Resolution of Vectors
- Motion in a Plane
- Projectile Motion
- Uniform Circular Motion (UCM)
- Equations of Motion in a Plane with Constant Acceleration
- Relative Velocity in Two Dimensions
