chapter 3 linear motion

Chapter 3: Linear Motion

Motion Is Relative

Motion is always described relative to a reference point, typically the surface of the Earth unless otherwise specified. Understanding relative motion is fundamental to analyzing how objects move in different frames of reference.

• Key Point: All motion is measured relative to something else.

• Example: A planet orbiting the sun is moving relative to the sun, not relative to the distant stars.

Speed

Speed is a scalar quantity representing the distance covered per unit of time. It does not include direction, only magnitude.

• Definition: Speed = distance / time

• SI Unit: meters per second (m/s)

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

• Example: If a car travels 100 km in 2 hours, its speed is 50 km/h.

Instantaneous Speed

Instantaneous speed is the speed at a specific moment in time, as shown by a speedometer.

• Example: If your speedometer reads 40 mph, your instantaneous speed is 40 mph at that instant.

Average Speed

Average speed is the total distance traveled divided by the total time taken. It is useful for planning trips and understanding overall motion.

• Formula: $\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}}$

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

Velocity

Velocity is a vector quantity, meaning it has both magnitude and direction. It describes how fast and in what direction an object moves.

• Definition: Velocity = speed with direction

• Symbol: v

• SI Unit: meters per second (m/s)

• Example: A car moving east at 100 km/h has a velocity of 100 km/h east.

• Comparison: Two cars moving at 100 km/h in opposite directions have the same speed but different velocities.

Constant Speed and Constant Velocity

Constant speed means the object moves at a steady rate. Constant velocity means both speed and direction remain unchanged, requiring a straight-line path.

• Constant Speed: No change in magnitude.

• Constant Velocity: No change in magnitude or direction.

Acceleration

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

• Definition: Acceleration = change in velocity / time

• Symbol: a

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

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

• Example: If a car speeds up from 13 m/s to 22 m/s in 10 s, its acceleration is $\frac{22 - 13}{10} = 0.9$ m/s2.

Types of Acceleration

• Accelerating: Acceleration and velocity are in the same direction.

• Decelerating: Acceleration and velocity are in opposite directions.

• Change in Direction: Even at constant speed, changing direction (e.g., circular motion) results in acceleration.

Galileo's Inclined Plane Experiments

Galileo studied acceleration using inclined planes, showing that steeper inclines produce greater acceleration. When the incline is vertical, acceleration equals that of free fall.

• Key Point: All objects fall with the same acceleration when air resistance is negligible.

Free Fall

Free fall occurs when an object is falling under the influence of gravity alone, with no air resistance. The acceleration due to gravity on Earth is approximately 9.8 m/s2 (often rounded to 10 m/s2 for calculations).

• On Earth: $g = 9.8$ m/s2

• On the Moon: $g_{moon} = 1.62$ m/s2

Free Fall—How Fast?

The speed of a freely-falling object starting from rest increases linearly with time.

• Formula: $\text{Speed} = g \times t$

• Example: After 3 seconds, speed is $9.8 \times 3 = 29.4$ m/s.

Free Fall—How Far?

The distance covered by a freely-falling object starting from rest increases with the square of the time.

• Formula: $\text{Distance} = \frac{1}{2} g t^2$

• Example: After 4 seconds, distance is $\frac{1}{2} \times 9.8 \times 4^2 = 78.4$ m.

Free Fall—Objects Thrown Straight Upward

When an object is thrown upward, it slows down due to gravity until it stops momentarily at its highest point, then accelerates downward. The upward and downward velocities at equal elevations are equal in magnitude but opposite in direction.

• Key Point: The acceleration due to gravity is always directed downward, regardless of the object's motion.

Velocity Vectors

Velocity vectors combine both magnitude and direction. When two velocity vectors are combined, such as an airplane flying north with a crosswind blowing east, the resultant velocity is found using vector addition.

• Example: An airplane flying north at 80 km/h with a 60 km/h crosswind east has a resultant velocity:

• Formula: $\text{Resultant velocity} = \sqrt{(80^2 + 60^2)} = 100$ km/h

Summary Table: Key Quantities in Linear Motion

Quantity	Definition	SI Unit	Formula
Speed	Distance per unit time	m/s	$\text{Speed} = \frac{\text{distance}}{\text{time}}$
Velocity	Speed with direction	m/s	$\text{Velocity} = \frac{\text{displacement}}{\text{time}}$
Acceleration	Change in velocity per unit time	m/s2	$\text{Acceleration} = \frac{\Delta v}{\Delta t}$
Free Fall Speed	Speed after time t	m/s	$\text{Speed} = g t$
Free Fall Distance	Distance after time t	m	$\text{Distance} = \frac{1}{2} g t^2$

Additional info:

• All equations use $g = 9.8$ m/s2 unless otherwise specified.

• For vector addition, use the Pythagorean theorem when vectors are perpendicular.