Describing Motion

Kinematics is the branch of physics that describes how objects move. Using calculus, we can precisely analyze position, velocity, and acceleration in both straight lines (one dimension) and on a plane (two dimensions).

Displacement, Velocity, and Acceleration

• Displacement measures change in position.
• Velocity is the rate of change of displacement: \( v = \frac{dx}{dt} \).
• Acceleration is the rate of change of velocity: \( a = \frac{dv}{dt} \).

When motion is not along a straight line, we use vectors to describe it. In two dimensions, position, velocity, and acceleration all have x and y components.

Calculus and Kinematics

• Integrating acceleration gives velocity.
• Integrating velocity gives displacement.

Real-World Example

The motion of a soccer ball kicked at an angle involves both horizontal and vertical motion, which can be analyzed using kinematic equations.

Typical Graphs

• Position vs. Time
• Velocity vs. Time
• Acceleration vs. Time

Understanding how these graphs relate helps make sense of motion in everyday life.