Displacement, velocity, and acceleration are fundamental concepts in physics that describe how objects move. These ideas form the basis for understanding more complex motion, allowing us to analyze everything from a ball's trajectory to a car's journey.
By simplifying objects to points and using mathematical formulas, we can calculate average and instantaneous values for these quantities. This approach helps us predict and explain various types of motion in the real world.
Change in object position
Object model simplification
Simplifies complex objects by treating them as a single point with properties (mass, charge) 📍
Ignores the size, shape, and internal configuration of the object
Allows for easier analysis of an object's motion and interactions
Displacement definition
Measures the change in an object's position from its initial to final location
Calculates the difference between the final position (x) and initial position (x0)
Uses the equation: Δx=x−x0
Represents the shortest distance between the starting and ending points, regardless of the actual path taken
Average velocity and acceleration
Calculation of averages
Determines the average values of velocity and acceleration over a specific time interval
Considers only the initial and final states of the object, not the intermediate values
Provides a simplified representation of an object's motion during the given time period
Average velocity formula
Calculates the average velocity by dividing the displacement (Δx) by the time interval (Δt)
Uses the equation: vavg =ΔtΔx
Represents the rate at which an object's position changes over time, in a particular direction
Does not provide information about the object's velocity at any specific instant
Average acceleration formula
Calculates the average acceleration by dividing the change in velocity (Δv) by the time interval (Δt)
Uses the equation: aavg =ΔtΔv
Represents the rate at which an object's velocity changes over time, in a particular direction
Does not provide information about the object's acceleration at any specific instant
Acceleration conditions
Occurs when either the magnitude or direction of an object's velocity changes
Happens when an object speeds up, slows down, or changes direction
Does not require a change in both magnitude and direction simultaneously
Can occur even if the object maintains a constant speed but changes direction (uniform circular motion) 🔄
Instantaneous kinematics
Limit of average values
Approaches the instantaneous value of a quantity as the time interval used to calculate the average approaches zero
Applies to position, velocity, and acceleration
Uses differentiation to determine instantaneous values from time-dependent functions
Relevant equations for instantaneous velocity:
v=dtdr (vector form)
vx=dtdx (component form)
Relevant equations for instantaneous acceleration:
a=dtdv (vector form)
ax=dtdvx (component form)
Time-dependent functions
Describes the position, velocity, and acceleration of an object as functions of time
Allows for the determination of instantaneous values at any given moment
Uses differentiation to find instantaneous velocity and acceleration from position functions
Uses integration to find displacement and change in velocity from velocity and acceleration functions, respectively 📈