Kinematics and Motion

Coordinate Systems and Units

Choosing and maintaining a consistent coordinate system is fundamental in physics problems. Proper unit conversion and significant digit handling ensure accuracy in calculations.

• Coordinate System: Select a coordinate system (e.g., x-y axes) and use it consistently throughout the problem.

• Unit Conversion: Multiply by ratios of equal quantities to cancel original units. For example, to convert meters to feet, multiply by 3.2808 feet / 1 meter.

• Significant Digits: Keep 4 significant digits in intermediate calculations; round final answers to 3 or 2 digits.

Average and Instantaneous Velocity

Velocity describes the rate of change of position. Average velocity is calculated over a time interval, while instantaneous velocity is the slope of the position vs. time graph at a specific moment.

• Average Velocity (1D): where is displacement and is the time interval.

• Instantaneous Velocity: The slope of the position vs. time graph at a given point.

Average and Instantaneous Acceleration

Acceleration measures the rate of change of velocity. Average acceleration is calculated over a time interval.

• Average Acceleration (1D): where is the change in velocity.

• Instantaneous Acceleration: The slope of the velocity vs. time graph at a specific moment.

Kinematics with Constant Velocity

When velocity is constant, displacement and position can be calculated directly.

• Displacement:

• Position as a Function of Time:

Kinematics with Constant Acceleration

For motion with constant acceleration, several key equations describe position, velocity, and displacement.

• Change in Velocity:

• Velocity as a Function of Time:

• Average Velocity:

• Vertical Displacement:

• Position as a Function of Time (Vertical):

• Velocity-Displacement Relation:

Projectile Motion

Projectile motion involves both horizontal and vertical components. The horizontal motion has constant velocity, while the vertical motion is affected by constant acceleration due to gravity.

• Horizontal Motion: direction has constant velocity.

• Vertical Motion: direction is "free fall" with constant acceleration (downward).

• Gravity:

• Finding Components: Use trigonometric functions (SOHCAHTOA) to resolve initial velocity into and components. Measure angle counterclockwise from the axis.

• Angle Convention: ; be careful if is negative.

• Example: A cannonball fired at an angle with speed has and .

Vectors and Vector Addition

In two or three dimensions, position, displacement, velocity, and acceleration are vectors. Vectors can be added graphically or algebraically.

• Graphical Addition: Tip-to-tail method.

• Algebraic Addition: Add and components separately.

Relative Motion

Relative motion describes how the velocity of one object appears from the perspective of another.

• Relative Velocity Formula: or

• Example: If a boat moves relative to the river, and the river moves relative to the shore, the boat's velocity relative to the shore is the sum of the two velocities.

Quadratic Formula

The quadratic formula is used to solve equations of the form .

• Quadratic Formula:

• Application: Used in kinematics to solve for time or position when equations are quadratic in form.

Unit Conversions

Unit conversions are essential for ensuring consistency in calculations.

Example: To convert 5 meters to feet: feet.