Kinematics in One and Two Dimensions

Displacement, Velocity, and Acceleration

Kinematics is the study of motion without considering its causes. It involves analyzing displacement, velocity, and acceleration in one or more dimensions.

• Displacement (): The change in position of an object. For straight-line motion, .

• Velocity (): The rate of change of displacement. Average velocity is .

• Acceleration (): The rate of change of velocity. .

• Area under velocity-time graph: Represents displacement. For constant velocity, area = velocity × time.

Example: If a velocity-time graph shows a constant velocity of for , the displacement is:

Instantaneous Velocity and Slope

The instantaneous velocity at a given time is the slope of the position vs. time graph at that point.

• Instantaneous velocity:

• Magnitude of velocity: For a given slope,

Example: If the slope of the position-time graph at is , then .

Kinematics in Two Dimensions

Projectile Motion

Projectile motion involves two-dimensional motion under constant acceleration due to gravity. The horizontal and vertical motions are independent.

• Horizontal motion: (no acceleration)

• Vertical motion: (acceleration )

• Maximum height: Occurs when vertical velocity is zero.

• Time to maximum height:

• Range:

Example: For a projectile launched at and angle :

• Horizontal velocity:

• Vertical velocity:

Statements about Projectile Motion

• At maximum height, vertical velocity is zero, but horizontal velocity remains unchanged.

• Speed at maximum height is equal to the horizontal component of velocity.

• Acceleration is always downward.

Solving for Time of Flight

To find the time a projectile spends in the air, use the vertical motion equation and solve for when returns to the initial height.

Example: If and :

• Set and solve for :

Relative Motion

Reference Frames and Relative Velocity

Relative velocity describes how the velocity of an object appears from different reference frames.

• Relative velocity equation:

• For objects moving on walkways or in boats, add velocities vectorially.

Example: Two people walking on moving walkways:

• Person A:

• Person B:

• Time to cross:

Worked Example: Projectile from a Building

Finding Speed and Time of Flight

When a projectile is launched from a height, use kinematic equations to find its speed and time to reach the ground.

• Vertical motion:

• Horizontal speed at impact:

• Time to reach a certain height: Solve for .

Example: For a projectile launched at and from height :

• Find when :

• Solve quadratic equation for .

Summary Table: Key Kinematic Equations

Additional info:

• Some context and equations have been expanded for clarity and completeness.

• Diagrams referenced in the original file have been described in text.