Linear Motion: Speed, Velocity, and Acceleration – Study Notes

Linear Motion

Introduction

Linear motion is a fundamental concept in physics that describes the movement of objects along a straight path. This topic covers the principles of speed, velocity, acceleration, and free fall, providing the foundation for understanding motion in one dimension.

Motion Is Relative

Reference Frames

The motion of objects is always described as relative to something else. Unless stated otherwise, we mean the motion relative to the surface of the Earth.

• Reference Frame: The perspective from which motion is measured (e.g., ground, moving vehicle).

• Example: A passenger walking inside a moving train has a velocity relative to the train and a different velocity relative to the ground.

Speed

Definition and Formula

Speed is the distance covered per unit of travel time. It is a scalar quantity, meaning it has magnitude but no direction.

• Formula: $\text{Speed} = \frac{\text{distance}}{\text{time}}$

• SI Unit: meters per second (m/s)

• Other Units: kilometers per hour (km/h), miles per hour (mph)

Instantaneous Speed

Instantaneous speed is the speed at a specific instant of time. It is what a speedometer shows at any moment.

• Example: If your speedometer reads 40 mph, you would cover 40 miles in one hour if you maintained that speed.

Average Speed

Average speed is the total distance covered divided by the total time interval.

• Formula: $\text{Average speed} = \frac{\text{total distance covered}}{\text{time interval}}$

• Application: Used in trip planning to estimate travel time.

• Example: Driving 30 km in 1 hour gives an average speed of 30 km/h. Driving 60 km in 2 hours also gives 30 km/h.

Velocity

Definition and Properties

Velocity is a vector quantity that includes both magnitude (speed) and direction. It answers "how fast" and "in what direction" an object is moving.

• Symbol: v

• Unit: m/s

• Example: A car travels at 60 mph to the north.

Comparing Velocities

• Two cars moving at 100 km/h in opposite directions do not have the same velocity, because their directions differ.

• Key Point: Velocity depends on both speed and direction.

Acceleration

Definition and Formula

Acceleration is the rate of change of velocity over time. It is a vector quantity, meaning it has both magnitude and direction.

• Formula: $a = \frac{\text{change in velocity}}{\text{time interval}}$

• Unit: meters per second squared (m/s2)

• Example: If a car speeds up from 13 m/s to 23 m/s in 10 s, its acceleration is $a = \frac{23 - 13}{10} = 1$ m/s2.

Types of Acceleration

• Positive Acceleration: Velocity and acceleration are in the same direction (speeding up).

• Negative Acceleration (Deceleration): Velocity and acceleration are in opposite directions (slowing down).

• Constant Acceleration: Acceleration remains the same over time (e.g., free fall).

Free Fall

Definition and Acceleration Due to Gravity

Free fall refers to motion under the influence of gravity only, with negligible air resistance. All objects in free fall near Earth's surface experience the same acceleration.

• Acceleration due to gravity (Earth): $g = 9.8$ m/s2 (often approximated as 10 m/s2)

• Acceleration due to gravity (Moon): $g_{\text{moon}} = 1.62$ m/s2

Equations of Motion for Free Fall

• Speed after time t (starting from rest): $v = gt$

• Speed with initial velocity: $v = at + v_0$

• Distance fallen after time t (starting from rest): $d = \frac{1}{2}gt^2$

• Distance with initial velocity: $d = v_0 t + \frac{1}{2}at^2$

Examples

• Speed after 1 second: $v = 10$ m/s (using $g = 10$ m/s2)

• Distance after 4 seconds: $d = \frac{1}{2} \times 10 \times 4^2 = 80$ m

Objects Thrown Upward

• When an object is thrown upward, it slows down at $g$ until it stops, then accelerates downward at $g$.

• At equal elevations, upward and downward velocities are equal in magnitude but opposite in direction.

Velocity Vectors

Combining Velocities

Velocity vectors can be combined to determine the resultant velocity of an object affected by multiple motions (e.g., an airplane flying in wind).

• Example: An airplane flying north at 80 km/h with a 60 km/h crosswind will have a resultant velocity determined by vector addition.

• Formula (Pythagorean theorem for perpendicular vectors): $v_{\text{resultant}} = \sqrt{v_1^2 + v_2^2}$

Summary Table: Key Quantities in Linear Motion

Quantity	Definition	Formula	SI Unit
Speed	Distance per unit time	$\text{Speed} = \frac{\text{distance}}{\text{time}}$	m/s
Velocity	Speed with direction	$\vec{v} = \frac{\text{displacement}}{\text{time}}$	m/s
Acceleration	Change in velocity per unit time	$a = \frac{\Delta v}{\Delta t}$	m/s2
Free Fall Acceleration	Acceleration due to gravity	$g = 9.8$ m/s2 (Earth)	m/s2

Additional info: Some equations and examples have been expanded for clarity and completeness. The summary table is inferred from the main concepts covered in the slides.