- \[\vec v_A\] = 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 \[\vec v_{BA}\], B is the observer, A is referenced.
Definitions [17]
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: 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.
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: 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.
Define angular velocity.
Angular velocity of a particle is the rate of change of angular displacement.
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.
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 'ΔtΔt' around time tt when the value of the time interval goes to zero is called instantaneous velocity.
Definition: Acceleration
Acceleration is defined as the rate of change of velocity with time.
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.
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.
Formulae [9]
Formula: Identity of Direction Cosines
The sum of squares of all direction cosines is always equal to 1:
cos2α + cos2β + cos2γ = 1
Formula: Three-Dimensional 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\[\hat k\]
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: 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 B relative to A
\[\vec{v}_{BA}=\vec{v}_B-\vec{v}_A\]
where:
Formula: Velocity of A relative to B
\[\vec{v}_{AB}=\vec{v}_A-\vec{v}_B\]
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.
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).
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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
