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

Linear Motion

Speed and Velocity

Linear motion refers to movement in a straight line and is a foundational concept in physics. Understanding the difference between speed and velocity is crucial for analyzing motion.

• Speed: The rate at which an object covers distance. It is a scalar quantity, meaning it has magnitude but no direction. Example: A car moving at 45 km/h.

• Velocity: The rate at which an object changes its position. It is a vector quantity, meaning it has both magnitude and direction. Example: Sue driving due north at 55 km/h.

• Formula for Speed: $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$

• Formula for Velocity: $\vec{v} = \frac{\Delta \vec{x}}{\Delta t}$

• Instantaneous Speed: The speed of an object at a particular moment in time.

• Constant Speed: When an object covers equal distances in equal intervals of time.

• Constant Velocity: Requires both constant speed and constant direction.

Additional info: Speed is always positive, while velocity can be positive or negative depending on direction.

Distance vs. Displacement

Distance and displacement are two ways to describe how far an object has moved.

• Distance: The total length of the path traveled by an object (scalar).

• Displacement: The straight-line distance from the starting point to the ending point, including direction (vector).

Acceleration

Acceleration is the rate at which velocity changes with time. It is a vector quantity and can be positive (speeding up) or negative (slowing down).

• Formula for Acceleration: $a = \frac{\Delta v}{\Delta t}$

• Units: $\text{m/s}^2$

• Example: A car increases its speed from 0 to 100 km/h in 10 seconds. $a = 10\ \text{m/s}^2$

• Free Fall Acceleration: Near Earth's surface, the acceleration due to gravity is approximately $10\ \text{m/s}^2$ downward.

• Direction of Acceleration: Acceleration points in the direction of the change in velocity.

Free Fall and Air Resistance

Objects in free fall experience constant acceleration due to gravity, unless air resistance is significant.

• Free Fall: Motion under the influence of gravity only.

• Acceleration due to Gravity: $g \approx 10\ \text{m/s}^2$

• Effect of Air Resistance: Air resistance decreases the acceleration of falling objects.

• Factors Affecting Air Resistance: Surface area and weight of the object.

Comparing Speed and Velocity

Speed is a scalar, while velocity is a vector. The difference involves direction.

• Speed: Only magnitude.

• Velocity: Magnitude and direction.

• Example: A car moving in a circular track at constant speed has changing velocity due to changing direction.

Galileo's Contributions

Galileo was the first to systematically study speed, velocity, and acceleration, laying the foundation for classical mechanics.

• Key Discoveries: All objects fall at the same rate in the absence of air resistance.

• Relationship: Distance traveled by a freely falling object is proportional to the square of the time elapsed.

Sample Table: Comparison of Speed, Velocity, and Acceleration

Quantity	Type	Formula	Units
Speed	Scalar	$\frac{\text{Distance}}{\text{Time}}$	m/s
Velocity	Vector	$\frac{\Delta \vec{x}}{\Delta t}$	m/s
Acceleration	Vector	$\frac{\Delta v}{\Delta t}$	m/s2

Key Examples and Applications

• Bus Example: If you walk at 10 km/h inside a bus moving at 100 km/h, your speed relative to the ground is 110 km/h.

• Bungee Jumper: During free fall, the speed increases by 10 m/s every second.

• Ball Thrown Upward: Loses 10 m/s of speed each second until it reaches the highest point, then gains 10 m/s each second on the way down.

Additional info: Acceleration is zero at the maximum height of a thrown ball, and negative when the ball is moving downward due to gravity.