Ch.03 – Linear Motion: Study Notes

Ch.03 – Linear Motion

Linear motion refers to any kind of motion in a straight line. This chapter covers the fundamental concepts of speed, velocity, acceleration, free fall, and velocity vectors, which are essential for understanding motion in physics.

3.1 Speed

Speed is a measure of how fast an object is moving, defined as the distance traveled per unit time. It is a scalar quantity, meaning it has magnitude but no direction.

• Instantaneous Speed: The speed at any instant of time. It is the value shown by a speedometer.

• Average Speed: The total distance traveled divided by the total time taken.

Formula for Average Speed:

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

Example: If a car travels 80 kilometers in 1 hour, the average speed is:

$\text{Average speed} = \frac{80\ \text{km}}{1\ \text{h}} = 80\ \text{km/h}$

3.2 Velocity

Velocity is speed in a specific direction. It is a vector quantity, meaning it has both magnitude and direction.

• Constant Velocity: Both speed and direction remain unchanged.

• Changing Velocity: Occurs if either the speed or the direction (or both) change.

Key Points:

• Two cars moving at the same speed but in opposite directions have different velocities.

• Constant speed does not mean constant velocity unless the direction is also constant.

Example: A car moving in a circle at constant speed has a changing velocity because its direction is constantly changing.

3.3 Acceleration

Acceleration is the rate at which velocity changes with respect to time. It can result from speeding up, slowing down, or changing direction.

• Acceleration is a vector quantity.

• Units: $\text{m/s}^2$ (meters per second squared)

Formula for Acceleration:

$\text{Acceleration} = \frac{\text{Change in velocity}}{\text{Change in time}}$

Example: If a car increases its velocity from 0 to 9 km/h in 1 second:

$\text{Acceleration} = \frac{9\ \text{km/h} - 0\ \text{km/h}}{1\ \text{s}} = 9\ \text{km/h/s}$

Note: A decrease in speed (deceleration) is also acceleration, but in the opposite direction.

3.4 Free Fall

Free fall describes the motion of an object under the influence of gravity alone, with no air resistance.

• All objects in free fall near Earth's surface accelerate downward at the same rate, regardless of their mass.

Acceleration due to gravity:

$g = 10\ \text{m/s}^2$

Equations for Free Fall:

• Velocity after time $t$: $v = gt$

• Distance fallen after time $t$: $d = \frac{1}{2}gt^2$

Example: If you drop a ball, after 2 seconds:

$\text{Velocity}: v = (10\ \text{m/s}^2)(2\ \text{s}) = 20\ \text{m/s}$ $\text{Distance}: d = \frac{1}{2}(10\ \text{m/s}^2)(2\ \text{s})^2 = 20\ \text{m}$

3.5 Velocity Vectors

Velocity vectors follow the rules of vector addition. When two velocities are at an angle, their resultant is found using vector addition methods.

• When moving in the same direction, add the velocities.

• When moving in opposite directions, subtract the velocities.

• When moving at an angle, use the Pythagorean theorem for perpendicular vectors.

Examples:

• A river flows east at 5 m/s and you swim north at 5 m/s. Your resultant velocity is:

$\text{Resultant velocity} = \sqrt{(5\ \text{m/s})^2 + (5\ \text{m/s})^2} = 7.1\ \text{m/s}$

• An airplane flying north at 150 km/h is caught in a 80 km/h crosswind eastward. The resultant velocity is:

$\text{Resultant velocity} = \sqrt{(150\ \text{km/h})^2 + (80\ \text{km/h})^2} = 170\ \text{km/h}$

Table: Free Fall Calculations

Time (s)	Velocity (m/s)	Distance (m)
0	0	0
1	10	5
2	20	20
3	30	45

Additional info: The table above summarizes the velocity and distance for a freely falling object at different times, using $g = 10\ \text{m/s}^2$.