Linear Motion: Speed, Velocity, and Acceleration (Chapter 3)

3.1 Speed

Definition of Speed

Speed is a measure of how fast an object moves, defined as the distance covered per unit of time. Galileo was the first to precisely define and measure speed.

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

• Units: The SI unit is meters per second (m/s). Other common units include kilometers per hour (km/h) and miles per hour (mph).

Instantaneous Speed vs. Average Speed

• Instantaneous Speed: The speed of an object at a specific moment (e.g., a car's speedometer reading).

• Average Speed: The total distance covered divided by the total time interval. $\text{Average Speed} = \frac{\text{Total Distance Covered}}{\text{Time Interval}}$

• Distance Calculation: $\text{Total Distance} = \text{Average Speed} \times \text{Time Interval}$

• Key Distinction: Average speed does not reflect variations in speed during a trip.

Motion is Relative

• Reference Frames: Motion is always described relative to a chosen frame of reference (e.g., Earth, Sun).

• Default Reference: Unless otherwise stated, speeds are measured relative to Earth's surface.

3.2 Velocity

Speed vs. Velocity

Velocity is a vector quantity that specifies both the speed and direction of an object's motion, while speed is a scalar (magnitude only).

• Speed (Scalar): Only magnitude (e.g., 60 km/h).

• Velocity (Vector): Magnitude and direction (e.g., 60 km/h North).

• Vector Quantity: Requires both magnitude and direction (e.g., velocity, force).

• Scalar Quantity: Requires only magnitude (e.g., speed, distance).

Constant Velocity

• Conditions: Constant speed and constant direction (straight-line motion at a steady rate).

Changing Velocity

• Triggers: Velocity changes if speed, direction, or both change.

• Example: A car moving at steady speed on a curved track has changing velocity due to changing direction.

Conceptual Examples

• Same Speed, Different Velocity: Two cars at 100 km/h, one east and one west, have the same speed but opposite velocities.

3.3 Acceleration

Definition of Acceleration

Acceleration is the rate at which velocity changes over time, including changes in speed, direction, or both.

• Formula: $\text{Acceleration} = \frac{\text{Change of Velocity}}{\text{Time Interval}} = \frac{\Delta v}{\Delta t}$

• Units: m/s2 or km/h·s (velocity units divided by time units).

Three Ways an Object Accelerates

• Increasing Speed: Pressing the gas pedal.

• Decreasing Speed (Deceleration): Applying brakes (negative acceleration).

• Changing Direction: Turning the steering wheel.

• Car Controls Analogy: Accelerator (increase speed), brake (decrease speed), steering wheel (change direction).

Straight-Line Motion & Velocity vs. Acceleration

• Straight-Line Motion: Acceleration is the rate of change of speed when direction is constant. $\text{Acceleration (straight line)} = \frac{\text{Change in Speed}}{\text{Time Interval}}$

• Zero Acceleration: Velocity is constant (not necessarily zero).

• High Constant Speed: An object at 400 km/h with no change in speed or direction has zero acceleration.

• Short Time Intervals: Small velocity changes over short times yield large accelerations.

Galileo’s Inclined Planes & Velocity Equations

• Idealized Cases: Neglecting air resistance and friction isolates fundamental motion principles.

• Constant Acceleration: Equal speed gained in equal time intervals (e.g., ball rolling down an incline).

• Velocity Equation (from rest): $v = at$

• Free Fall Acceleration: On a vertical incline, acceleration is due to gravity ($g$), approximately 10 m/s2.

3.4 Free Fall

Concept of Free Fall

• Definition: Motion under gravity alone, without air resistance.

• Acceleration Due to Gravity ($g$): $g \approx 9.8\ \text{m/s}^2$ (often rounded to 10 m/s2 for calculations).

• Universality: In a vacuum, all objects fall with the same acceleration regardless of mass or shape.

How Fast (Instantaneous Velocity)

• Formula (from rest): $v = gt$

• Key Characteristics: Velocity increases by 10 m/s each second (if $g = 10\ \text{m/s}^2$).

• Upward Motion: Objects thrown up lose 10 m/s each second until velocity is zero, then gain 10 m/s downward per second.

• Symmetry: At equal heights, speed magnitude is the same going up or down (directions are opposite).

How Far (Distance Fallen)

• Formula (from rest): $d = \frac{1}{2}gt^2$

• Key Distinction: "How fast" refers to velocity ($v$), "how far" to distance ($d$).

• First Second Example: After 1 s, $v = 10\ \text{m/s}$, but $d = 5\ \text{m}$ (average speed is 5 m/s).

Physics of "Hang Time"

• Definition: The total time an athlete is airborne during a jump.

• Jumping Mechanics: Upward speed decreases at $g$; time up equals time down.

• Formula: $t = \sqrt{\frac{2d}{g}}$ (for half the total hang time)

• Typical Values: Exceptional vertical leaps rarely exceed 0.6 m (2 ft) with hang times around 1 second.

3.5 Velocity Vectors

Speed vs. Velocity

• Speed (Scalar): Magnitude only (e.g., 100 km/h).

• Velocity (Vector): Magnitude and direction (e.g., 100 km/h North).

Acceleration as a Vector

• Direction of Acceleration: Always in the direction of the net force.

• Relation to Velocity:

  • Same direction: Object speeds up.

  • Opposite direction: Object slows down.

  • At an angle: Object changes direction.

Combining Velocity Vectors

• Resultant Motion: The vector sum of independent velocities gives the actual path and speed.

• Perpendicular Vectors (Pythagorean Theorem): $\text{Resultant Speed} = \sqrt{v_1^2 + v_2^2}$

• Example (Airplane in Crosswind): Airplane at 120 km/h with a 90 km/h crosswind: $\text{Resultant Speed} = \sqrt{120^2 + 90^2} = 150\ \text{km/h}$

• Example (Swimming Across a River): Swimmer at 4 m/s in a 4 m/s current: $\text{Resultant Speed} = \sqrt{4^2 + 4^2} = 5.7\ \text{m/s}$