Chapter 2: One Dimensional Kinematics (Straight Line Motion)

# Introduction

Kinematics is the branch of physics that studies the motion of objects without considering the forces causing the motion.

One-dimensional kinematics deals with motion along a straight line, considering only one direction.

# 2.1: Displacements

**Kinematics**Deals with the concepts that are needs to

__describe motion__. without any reference to forces.__How things move__**Straight line motion**

**Dynamics**deals with the effect that forces have on motionKinematics and dynamics from the branch of physics we know as

**mechanics.****Displacement visual:**

### Variable Meanings:

${x}_{0}$ = initial position

$x$ = final position

$\Delta$ = final - initial

Change in

$\Delta x=x-x_{0}$

Arrows over variable represent vectors

# 2.2: Speed and Velocity

### Average speed

**Average speed**is the distance traveled divided by the time required to cover the distance.Average speed = $\dfrac{Distance}{ElapsedTime}$

Elapsed time = $\Delta t=t-t_{0}$

**Distance**(m)A

**scalar unit**measure of the**distance**an object moves as measured along the path followed.

Scalar quantity

SI units for speed: $\dfrac{m}{s}$

### Average velocity

**Average velocity**is the displacement divided by the elapsed time.Average velocity = $\dfrac{Displacement}{ElapsedTime}$ = $\dfrac{\Delta \overrightarrow{x}}{\Delta t}$

$\Delta x=x-x_{0}$

$\Delta t=t-t_{0}$

If average velocity is

**positive**, the object is moving in the**positive**direction.If average velocity is

**negative**, the object is moving in the**negative**direction.If average velocity is

**zero**, the object is**not moving**.**Displacement**(m)A

**vector**representing the**change in position**of an object, drawn from the**initial**to the**final**position.

Vector quantity

SI unit for velocity: $\dfrac{m}{s}$

### Instantaneous velocity

The instantaneous velocity indicates how fast an object is moving and the direction of motion at each instant of time.

The rate of change of the displacement at a particular instant.

$\overrightarrow{v}=\lim _{\Delta t-0}\dfrac{\overrightarrow{\Delta x}}{\Delta t}$

Vector quantity

SI unit for velocity: $\dfrac{m}{s}$

# 2.3: Acceleration

**Acceleration**is any**change in velocity**for an object divided by the time interval over which the change occurs.The notion of acceleration emerges when a change in velocity is combined with the time during which the change occurs.

The change in the

**velocity**of an object divided by the elapsed**time**.Definition of average acceleration:

$\dfrac{}{\overrightarrow{a}}=\dfrac{\overrightarrow{v}-\overrightarrow{v}_{0}}{t-t_{0}}=\dfrac{\Delta \overrightarrow{v}}{\Delta t}$

If average acceleration is positive, the speed is increasing with time.

If average acceleration is negative, the speed is decreasing with time.

If average acceleration is zero, it is at constant speed.

Vector quantity

SI unit for velocity: $\dfrac{m}{s^{2}}$

# 2.4: Equations of Kinematics for Constant Acceleration

Develop useful equations assuming:

Acceleration is constant

Start motion at origin

Start clock at time = 0

**Five kinematic variables**Meaning

Variable

Units

Displacement

$x$

m

Acceleration (constant)

$a$

$\dfrac{m}{s^{2}}$

Final velocity (at time t)

$v$

$\dfrac{m}{s}$

Initial velocity

$v_{0}$

$\dfrac{m}{s}$

Elapsed time

$t$

s

**Kinematic equations**$v=v_{0}+at$

$x=\dfrac{1}{2}\left( v_{0}+v\right) t$

$v^{2}=v_{0}^{2}+2ax$

$x=v_{0}t+\dfrac{1}{2}at^{2}$

# 2.5: Applications of the Equations of Kinematics

### Reasoning Strategy

Make a drawing.

Decide which directions are positive and negative.

Write down the value that are given for any of the five kinematic variable.

Determined the desired variable.

Each equation has only 4 variables.

If we know 3, we can solve for the rest.

Find the appropriate kinematic equation to apply.

Solve.

# 2.6: Freely Falling Bodies

**Free Fall**- An idealized vertical motion in which air resistance is.__IGNORED__All bodies at the same location above the Earth fall vertically with the same acceleration.

This motion of free fall and the acceleration of a freely falling body is called

**acceleration due to gravity**.Acceleration due to gravity is always constant and always faces downwards.

The

**ONLY**force acting on the object is the**CONSTANT FORCE OF**.__GRAVITY__Causes constant

**acceleration of -9.80 m/s²**a =-9.80 m/s²

g = 9.80 m/s²

g is ALWAYS positive.

If a ball is thrown in the air, gravity is pulling it down.

# Graphical Analysis of Velocity and Acceleration-Velocity

Slope = $\dfrac{rise}{run}=\dfrac{\Delta x}{\Delta t}=\overline{v}$

## ON A DISPLACEMENT-TIME GRAPH

**SLOPE**equals**VELOCITY**(rise/run)=m/s)**Y-INTERCEPT**equals the**INTIAL DISPLACEMENT****STRAIGHT LINES**imply__zero velocity__(constant velocity)**CURVED LINES**imply__non-uniform__acceleration**PARABOLIC LINES**imply__constant__acceleration**AVERAGE VELOCITY**is the__slope__of the__straight line connecting the endpoints__of a curve**INSTANTANEOUS VELOCITY**is the__slope__of the__line tangent__to a curve at any point**POSITIVE SLOPE**implies motion in the__positive direction__**NEGATIVE SLOPE**implies motion in the__negative direction__**ZERO SLOPE**implies__state of rest__

## ON A VELOCITY-TIME GRAPH

**SLOPE**equals**ACCELERATION**(rise/run = m/s / s=m/sa²)**Y-INTERCEPT**equals the**INITIAL VELOCITY****HORIZONTAL LINES**imply__zero acceleration__**STRAIGHT LINES**imply__constant acceleration__**CURVED LINES**imply__non-uniform acceleration__**AVERAGE ACCELERATION**is the__slope__of the__straight line connecting the endpoints__of a curve**INSTANTANEOUS ACCELERATION**is the__slope__of the__line tangent__to a curve at any point**POSITIVE SLOPE**implies an__increase in velocity__in the__positive direction__**NEGATIVE SLOPE**implies an__increase__in velocity in the__negative direction__**ZERO SLOPE**implies__motion__with__constant velocity__

## ON AN ACCELERATION-TIME GRAPH

**SLOPE**is meaningless, but__zero slope__(a horizontal line) implies motion with zero**or**constant acceleration**Y-INTERCEPTS**equals the__initial acceleration__The

**area under the curve**__equals__the**change in velocity**(m/s² x s = m/s)

# Extra Notes

## Constant Acceleration

**Constant acceleration**refers to a situation where an object's velocity changes by the same amount in each unit of time. In other words, the object's acceleration remains constant over time. This can be represented by a straight line on a velocity-time graph.If something is

**speeding up**in a given direction, the**acceleration must be in that direction**.If something is

**slowing down**, the**acceleration is opposite to the direction of motion**.

**Example: **A car moving along a straight road with constant acceleration can be exemplified by a scenario where its velocity increases by 10 m/s every second. Initially, the car is at rest, but after 1 second, its velocity becomes 10 m/s. After 2 seconds, it reaches 20 m/s, and so on. This consistent increase in velocity over time demonstrates constant acceleration.

## Freely Falling Bodies

In absence of air resistance, all bodies near the surface of the Earth fall vertically with the same

**constant acceleration**(because of constant force).This is called the

**acceleration due to gravity**(gravity being the force that causes the acceleration.Represented by symbol

**g**.

Force of gravity is

**ALWAYS**present.Acceleration is

**ALWAYS**present,**ALWAYS**the same value, and**ALWAYS**directed downward (toward center of Earth, so its negative if pointing down).**g = -9.80 m/s²**g may be exchanged with a = -9.80 m/s²

Since the force of air resistance increases with speed, sometimes objects will reach a maximum “terminal velocity” and stop accelerating when at some point

**net acceleration = 0**

## Terminal Velocity (v+)

**Terminal velocity**is the maximum velocity that an object can reach when falling through a fluid, such as air or water. It occurs when the force of gravity pulling the object downward is balanced by the drag force exerted by the fluid. At terminal velocity, the object no longer accelerates and falls at a constant speed. The value of terminal velocity depends on the object's mass, shape, and the density of the fluid it is falling through.**EQUATION FOR VELOCITY**Velocity = gravity(time)

v = gt

v=g(t) is a vacuum, so v continually increases

## Key Concepts

Displacement:

The change in position of an object.

It is a vector quantity, having both magnitude and direction.

Symbolized by Δx.

Velocity:

The rate of change of displacement.

It is a vector quantity, having both magnitude and direction.

Symbolized by v.

Average velocity (v_avg) is calculated as the displacement divided by the time taken.

Speed:

The rate of change of distance traveled.

It is a scalar quantity, having only magnitude.

Symbolized by s.

Average speed (s_avg) is calculated as the total distance traveled divided by the time taken.

Acceleration:

The rate of change of velocity.

It is a vector quantity, having both magnitude and direction.

Symbolized by a.

Average acceleration (a_avg) is calculated as the change in velocity divided by the time taken.

Equations of Motion:

The equations that relate displacement, velocity, acceleration, and time.

For constant acceleration, the following equations hold:

v = u + at (equation 1)

s = ut + (1/2)at^2 (equation 2)

v^2 = u^2 + 2as (equation 3)

s = ((u + v)/2)t (equation 4)

Where u is the initial velocity, v is the final velocity, t is the time, and a is the acceleration.

Graphical Representations:

Displacement-time graph:

The slope of the graph represents velocity.

Velocity-time graph:

The area under the graph represents displacement.

Acceleration-time graph:

The slope of the graph represents acceleration.

## Conclusion

One-dimensional kinematics focuses on the motion of objects along a straight line.

Displacement, velocity, speed, and acceleration are fundamental concepts in kinematics.

Equations of motion and graphical representations help analyze and understand the motion of objects.

# Chapter 2: One Dimensional Kinematics (Straight Line Motion)

# Introduction

Kinematics is the branch of physics that studies the motion of objects without considering the forces causing the motion.

One-dimensional kinematics deals with motion along a straight line, considering only one direction.

# 2.1: Displacements

**Kinematics**Deals with the concepts that are needs to

__describe motion__. without any reference to forces.__How things move__**Straight line motion**

**Dynamics**deals with the effect that forces have on motionKinematics and dynamics from the branch of physics we know as

**mechanics.****Displacement visual:**

### Variable Meanings:

${x}_{0}$ = initial position

$x$ = final position

$\Delta$ = final - initial

Change in

$\Delta x=x-x_{0}$

Arrows over variable represent vectors

# 2.2: Speed and Velocity

### Average speed

**Average speed**is the distance traveled divided by the time required to cover the distance.Average speed = $\dfrac{Distance}{ElapsedTime}$

Elapsed time = $\Delta t=t-t_{0}$

**Distance**(m)A

**scalar unit**measure of the**distance**an object moves as measured along the path followed.

Scalar quantity

SI units for speed: $\dfrac{m}{s}$

### Average velocity

**Average velocity**is the displacement divided by the elapsed time.Average velocity = $\dfrac{Displacement}{ElapsedTime}$ = $\dfrac{\Delta \overrightarrow{x}}{\Delta t}$

$\Delta x=x-x_{0}$

$\Delta t=t-t_{0}$

If average velocity is

**positive**, the object is moving in the**positive**direction.If average velocity is

**negative**, the object is moving in the**negative**direction.If average velocity is

**zero**, the object is**not moving**.**Displacement**(m)A

**vector**representing the**change in position**of an object, drawn from the**initial**to the**final**position.

Vector quantity

SI unit for velocity: $\dfrac{m}{s}$

### Instantaneous velocity

The instantaneous velocity indicates how fast an object is moving and the direction of motion at each instant of time.

The rate of change of the displacement at a particular instant.

$\overrightarrow{v}=\lim _{\Delta t-0}\dfrac{\overrightarrow{\Delta x}}{\Delta t}$

Vector quantity

SI unit for velocity: $\dfrac{m}{s}$

# 2.3: Acceleration

**Acceleration**is any**change in velocity**for an object divided by the time interval over which the change occurs.The notion of acceleration emerges when a change in velocity is combined with the time during which the change occurs.

The change in the

**velocity**of an object divided by the elapsed**time**.Definition of average acceleration:

$\dfrac{}{\overrightarrow{a}}=\dfrac{\overrightarrow{v}-\overrightarrow{v}_{0}}{t-t_{0}}=\dfrac{\Delta \overrightarrow{v}}{\Delta t}$

If average acceleration is positive, the speed is increasing with time.

If average acceleration is negative, the speed is decreasing with time.

If average acceleration is zero, it is at constant speed.

Vector quantity

SI unit for velocity: $\dfrac{m}{s^{2}}$

# 2.4: Equations of Kinematics for Constant Acceleration

Develop useful equations assuming:

Acceleration is constant

Start motion at origin

Start clock at time = 0

**Five kinematic variables**Meaning

Variable

Units

Displacement

$x$

m

Acceleration (constant)

$a$

$\dfrac{m}{s^{2}}$

Final velocity (at time t)

$v$

$\dfrac{m}{s}$

Initial velocity

$v_{0}$

$\dfrac{m}{s}$

Elapsed time

$t$

s

**Kinematic equations**$v=v_{0}+at$

$x=\dfrac{1}{2}\left( v_{0}+v\right) t$

$v^{2}=v_{0}^{2}+2ax$

$x=v_{0}t+\dfrac{1}{2}at^{2}$

# 2.5: Applications of the Equations of Kinematics

### Reasoning Strategy

Make a drawing.

Decide which directions are positive and negative.

Write down the value that are given for any of the five kinematic variable.

Determined the desired variable.

Each equation has only 4 variables.

If we know 3, we can solve for the rest.

Find the appropriate kinematic equation to apply.

Solve.

# 2.6: Freely Falling Bodies

**Free Fall**- An idealized vertical motion in which air resistance is.__IGNORED__All bodies at the same location above the Earth fall vertically with the same acceleration.

This motion of free fall and the acceleration of a freely falling body is called

**acceleration due to gravity**.Acceleration due to gravity is always constant and always faces downwards.

The

**ONLY**force acting on the object is the**CONSTANT FORCE OF**.__GRAVITY__Causes constant

**acceleration of -9.80 m/s²**a =-9.80 m/s²

g = 9.80 m/s²

g is ALWAYS positive.

If a ball is thrown in the air, gravity is pulling it down.

# Graphical Analysis of Velocity and Acceleration-Velocity

Slope = $\dfrac{rise}{run}=\dfrac{\Delta x}{\Delta t}=\overline{v}$

## ON A DISPLACEMENT-TIME GRAPH

**SLOPE**equals**VELOCITY**(rise/run)=m/s)**Y-INTERCEPT**equals the**INTIAL DISPLACEMENT****STRAIGHT LINES**imply__zero velocity__(constant velocity)**CURVED LINES**imply__non-uniform__acceleration**PARABOLIC LINES**imply__constant__acceleration**AVERAGE VELOCITY**is the__slope__of the__straight line connecting the endpoints__of a curve**INSTANTANEOUS VELOCITY**is the__slope__of the__line tangent__to a curve at any point**POSITIVE SLOPE**implies motion in the__positive direction__**NEGATIVE SLOPE**implies motion in the__negative direction__**ZERO SLOPE**implies__state of rest__

## ON A VELOCITY-TIME GRAPH

**SLOPE**equals**ACCELERATION**(rise/run = m/s / s=m/sa²)**Y-INTERCEPT**equals the**INITIAL VELOCITY****HORIZONTAL LINES**imply__zero acceleration__**STRAIGHT LINES**imply__constant acceleration__**CURVED LINES**imply__non-uniform acceleration__**AVERAGE ACCELERATION**is the__slope__of the__straight line connecting the endpoints__of a curve**INSTANTANEOUS ACCELERATION**is the__slope__of the__line tangent__to a curve at any point**POSITIVE SLOPE**implies an__increase in velocity__in the__positive direction__**NEGATIVE SLOPE**implies an__increase__in velocity in the__negative direction__**ZERO SLOPE**implies__motion__with__constant velocity__

## ON AN ACCELERATION-TIME GRAPH

**SLOPE**is meaningless, but__zero slope__(a horizontal line) implies motion with zero**or**constant acceleration**Y-INTERCEPTS**equals the__initial acceleration__The

**area under the curve**__equals__the**change in velocity**(m/s² x s = m/s)

# Extra Notes

## Constant Acceleration

**Constant acceleration**refers to a situation where an object's velocity changes by the same amount in each unit of time. In other words, the object's acceleration remains constant over time. This can be represented by a straight line on a velocity-time graph.If something is

**speeding up**in a given direction, the**acceleration must be in that direction**.If something is

**slowing down**, the**acceleration is opposite to the direction of motion**.

**Example: **A car moving along a straight road with constant acceleration can be exemplified by a scenario where its velocity increases by 10 m/s every second. Initially, the car is at rest, but after 1 second, its velocity becomes 10 m/s. After 2 seconds, it reaches 20 m/s, and so on. This consistent increase in velocity over time demonstrates constant acceleration.

## Freely Falling Bodies

In absence of air resistance, all bodies near the surface of the Earth fall vertically with the same

**constant acceleration**(because of constant force).This is called the

**acceleration due to gravity**(gravity being the force that causes the acceleration.Represented by symbol

**g**.

Force of gravity is

**ALWAYS**present.Acceleration is

**ALWAYS**present,**ALWAYS**the same value, and**ALWAYS**directed downward (toward center of Earth, so its negative if pointing down).**g = -9.80 m/s²**g may be exchanged with a = -9.80 m/s²

Since the force of air resistance increases with speed, sometimes objects will reach a maximum “terminal velocity” and stop accelerating when at some point

**net acceleration = 0**

## Terminal Velocity (v+)

**Terminal velocity**is the maximum velocity that an object can reach when falling through a fluid, such as air or water. It occurs when the force of gravity pulling the object downward is balanced by the drag force exerted by the fluid. At terminal velocity, the object no longer accelerates and falls at a constant speed. The value of terminal velocity depends on the object's mass, shape, and the density of the fluid it is falling through.**EQUATION FOR VELOCITY**Velocity = gravity(time)

v = gt

v=g(t) is a vacuum, so v continually increases

## Key Concepts

Displacement:

The change in position of an object.

It is a vector quantity, having both magnitude and direction.

Symbolized by Δx.

Velocity:

The rate of change of displacement.

It is a vector quantity, having both magnitude and direction.

Symbolized by v.

Average velocity (v_avg) is calculated as the displacement divided by the time taken.

Speed:

The rate of change of distance traveled.

It is a scalar quantity, having only magnitude.

Symbolized by s.

Average speed (s_avg) is calculated as the total distance traveled divided by the time taken.

Acceleration:

The rate of change of velocity.

It is a vector quantity, having both magnitude and direction.

Symbolized by a.

Average acceleration (a_avg) is calculated as the change in velocity divided by the time taken.

Equations of Motion:

The equations that relate displacement, velocity, acceleration, and time.

For constant acceleration, the following equations hold:

v = u + at (equation 1)

s = ut + (1/2)at^2 (equation 2)

v^2 = u^2 + 2as (equation 3)

s = ((u + v)/2)t (equation 4)

Where u is the initial velocity, v is the final velocity, t is the time, and a is the acceleration.

Graphical Representations:

Displacement-time graph:

The slope of the graph represents velocity.

Velocity-time graph:

The area under the graph represents displacement.

Acceleration-time graph:

The slope of the graph represents acceleration.

## Conclusion

One-dimensional kinematics focuses on the motion of objects along a straight line.

Displacement, velocity, speed, and acceleration are fundamental concepts in kinematics.

Equations of motion and graphical representations help analyze and understand the motion of objects.