- \[\vec v_A\] = 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 [44]
Definition: Motion
"Motion is a change in the position of an object with time."
Define the following term:
Free fall
A path of free fall is the term used to describe the movement of an object solely under the influence of gravity.
Definition: Acceleration
The rate of change of velocity of an object with time is called Acceleration.
Definition: Instantaneous Acceleration
The limiting value of the average acceleration of an object over a small time interval Δt around time t, when the value of the time interval approaches zero, is called Instantaneous Acceleration.
Definition: Relative Velocity
The velocity of one object as observed by another object is called Relative Velocity.
Definition: Average Speed
The total path length (distance) travelled by an object during a time interval, divided by that time interval, is called Average Speed.
Definition: Instantaneous Velocity
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 approaches zero, is called Instantaneous Velocity.
Definition: Instantaneous Speed
The limiting value of the average speed of an object over a small time interval Δt around time t, when the value of the time interval approaches zero, is called Instantaneous Speed.
Definition: Rectilinear Motion
The motion of an object in which the position of a particle varies only in terms of distance along a straight line is called Rectilinear Motion.
Definition: Displacement
The shortest straight-line distance between an object's initial and final positions, represented as the difference between the position vectors Δx = x2 − x1, is called Displacement.
Definition: Path Length
The total distance travelled by an object along its actual path, regardless of the direction of motion, is called Path Length.
Definition: Average Velocity
The limiting value of the total displacement Δx of an object during a time interval Δt, divided by that time interval, is called Average Velocity.
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: 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."
Definition: Average Speed
"total path length travelled during the time interval over which average speed is being calculated, divided by that time interval."
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.
Definition: Instantaneous Speed
Instantaneous speed is simply the speed of an object at a single, specific moment in time (t).
Definition: Acceleration
Acceleration is defined as the rate of change of velocity with time.
Definition: Relative Velocity
Relative velocity is the velocity of one object as measured from another moving object's perspective.
Let:
- vA = velocity of object A (relative to ground/Earth)
- vB = velocity of object B (relative to ground/Earth)
- vAB = velocity of A relative to B (what B observes about A's motion)
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).
Definition: Instantaneous Velocity in Two-Dimensional Motion
The velocity at an exact moment—drawn as the tangent to the path.
Definition: Average Velocity in Two-Dimensional Motion
Total displacement divided by elapsed time.
Definition: Instantaneous Acceleration in plane
Instantaneous Acceleration (\[\vec a\]): The acceleration of an object at a specific instant of time. It is the limit of average acceleration as the time interval approaches zero, or the rate of change of velocity with respect to time.
Definition: Average acceleration in plane
Average Acceleration (\[\vec a_{av}\]): The change in velocity divided by the time interval over which that change occurs. It represents the acceleration over a specific time period between two moments.
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 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: 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: 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: 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.
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: 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: Time Period (T)
The time taken by a particle performing uniform circular motion to complete one revolution is called time period.
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 Velocity (ω)
The rate of change of angular displacement of a body undergoing circular motion is called angular velocity.
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: 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.
Define angular velocity.
Angular velocity of a particle is the rate of change of angular displacement.
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 Speed
Angular speed (ω) is the angle described by the radius vector per unit time.
Definition: Linear Speed
This is the familiar speed (distance/time). In one period (T), the distance travelled is the circumference of the circle, 2πr.
Definition: Period
This is the time it takes for the object to complete one full lap (one revolution). Its unit is seconds (s).
Definition: Radius Vector
It is a vector that points from the center of the circle (the origin) out to the position of the particle.
- Magnitude: Its length is simply the radius, r.
- Key Insight: As the object moves in UCM, its radius vector sweeps out equal angles in equal amounts of time.
Definition: Centripetal Force
A force that acts on any object moving along a circle and is directed towards the centre of the circle. When this force stops acting, the object flies off along a straight line (tangent to the circle) in the direction of its velocity at that instant.
Formulae [18]
Formula: Displacement
If the position at time t₁ is x₁ and at time t₂ is x₂, then
Displacement \[\vec s\] = \[\vec x_2\] - \[\vec x_1\]
Formula
\[\vec{v}_{\mathrm{av}}=\frac{\vec{x}_2-\vec{x}_1}{t_2-t_1}\]
- vav : average velocity.
- x2 : final position vector.
- x1 : initial position vector.
- t2 : final time
- t1 : initial time
Dimensions: [L1M0T−1]
Formula: Average Speed
Average Speed = vav = \[\frac{\text{path length}}{\text{time interval}}\]
Formula: Instantaneous velocity
\[\vec{\mathrm{v}}=\lim_{\Delta t\to0}\left(\frac{\Delta\vec{x}}{\Delta t}\right)=\frac{d\vec{x}}{dt}\]
Formula: Instantaneous Speed
To calculate instantaneous speed, we look at the average speed () over a very, very short time interval (Δt). It is defined as the limiting value of the average speed as the time interval (Δt) approaches zero.
Instantaneous Speed = \[\operatorname*{lim}_{\Delta t\to0}\frac{\mathrm{Distance}}{\Delta t}\]
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
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.
Formula: Relative Velocity
vAB = vA - vB
vBA = vB - vA = -vAB
Key relationship: vAB = -vBA
Formula: Instantaneous Velocity in Two-Dimensional Motion
\[\vec{v}=\lim_{\Delta t\to0}\frac{\Delta\vec{r}}{\Delta t}=\frac{d\vec{r}}{dt}=\frac{dx}{dt}\hat{i}+\frac{dy}{dt}\hat{j}\]
Magnitude & Direction:
v = \[\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2},\quad\theta=\tan^{-1}\left(\frac{dy/dt}{dx/dt}\right)\]
Curve of motion with a tangent line at P, and component arrows showing vx and vy.)
Instantaneous velocity
Formula: Average Velocity in Two-Dimensional Motion
\[\vec{v}_\mathrm{avg}=\frac{\Delta\vec{r}}{\Delta t}=\left(\frac{x_2-x_1}{t_2-t_1}\right)\hat{i}+\left(\frac{y_2-y_1}{t_2-t_1}\right)\hat{j}\]
Components:
- vavg,x = \[\frac{x_2-x_1}{t_2-t_1}\]
- vavg,y = \[\frac{y_2-y_1}{t_2-t_1}\]
Magnitude & Direction:
\[v_{\mathrm{avg}}=\sqrt{v_x^2+v_y^2},\quad\theta=\tan^{-1}\left(\frac{v_y}{v_x}\right)\]
Formula: Instantaneous Acceleration in Two-Dimensional plane
When the time interval becomes extremely small (approaches zero), the average acceleration becomes the instantaneous acceleration — i.e., acceleration at a specific instant.
\[\vec{a}=\lim_{\Delta t\to0}\frac{\Delta\vec{v}}{\Delta t}=\frac{d\vec{v}}{dt}\]
Or in components:
\[\vec{a}=\left(\frac{dv_x}{dt}\right)\hat{i}+\left(\frac{dv_y}{dt}\right)\hat{j}\]
Since \[v_x=\frac{dx}{dt}\mathrm{~and~}v_y=\frac{dy}{dt}\mathrm{:}\]
\[\vec{a}=\left(\frac{d^2x}{dt^2}\right)\hat{i}+\left(\frac{d^2y}{dt^2}\right)\hat{j}\]
Magnitude:
a = \[\sqrt{\left(\frac{d^2x}{dt^2}\right)^2+\left(\frac{d^2y}{dt^2}\right)^2}\]
Direction:
\[\tan\theta=\frac{dv_y/dt}{dv_x/dt}=\frac{dv_y}{dv_x}\]
Formula: Average Acceleration in Two-Dimensional plane
The average acceleration between t1 and t2 is
\[\vec{a}_{av}=\frac{\vec{v_2}-\vec{v_1}}{t_2-t_1}\]
That means: “Average acceleration = Change in velocity ÷ Time interval”
In component form:
\[\vec{a}_{av}=\left(\frac{v_{2x}-v_{1x}}{t_2-t_1}\right)\hat{i}+\left(\frac{v_{2y}-v_{1y}}{t_2-t_1}\right)\hat{j}\]
Magnitude:
\[a_{av}=\sqrt{(a_{av,x})^2+(a_{av,y})^2}\]
Direction:
\[\tan\theta=\frac{a_{av,y}}{a_{av,x}}\]
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: Linear Speed
v = \[\frac {Distance}{Time}\] = \[\frac {2πr}{T}\] (Unit: m/s)
Formula: Angular Speed
ω = \[\frac {Angle Swept}{Time}\] (Unit: radian/s)
Formula: Centripetal Force
F = \[m\omega^{2}r=\frac{mv^{2}}{r}\] = mωv
where:
- F = Centripetal force (in Newtons)
- m = Mass of the object (in kg)
- ω = Angular speed (in rad/s)
- r = Radius of the circular path (in m)
- v = Linear speed or tangential velocity (in m/s)
Formula: Centripetal Acceleration
a = ω2r = \[\frac {v^2}{r}\] = ωv
where:
-
a = Centripetal acceleration (in m/s²)
Key Points
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]
- Concept of Motion
- Rectilinear Motion
- Displacement
- Path Length
- Average Velocity
- Average Speed
- Instantaneous Velocity
- Instantaneous Speed
- Acceleration in Linear Motion
- Relative Velocity
- Motion in Two Dimensions - Motion in a Plane
- Average and Instantaneous Velocities
- Acceleration in a Plane
- Equations of Motion in a Plane with Constant Acceleration
- Relative Velocity in Two Dimensions
- Projectile Motion
- Uniform Circular Motion (UCM)
- Key Parameters of Circular Motion
- Centripetal Acceleration
- Conical Pendulum
