Representing motion

Motion can be described through motion diagrams, figures, graphs, equations, and narrative descriptions. Each representation highlights different aspects of the same physical situation. For constant acceleration, three kinematic equations relate position, velocity, acceleration, and time. Choosing the right equation depends on which variable is unknown and which three are given.

Key Definition Near Earth's surface, free-fall acceleration is approximately 10 m/s² directed downward, and it does not depend on the mass of the object.

Kinematic Equations for Constant Acceleration

v = v₀ + at x = x₀ + v₀t + ½at² v² = v₀² + 2a(x − x₀)

Graphs

• The slope of a position–time graph gives the instantaneous velocity.
• The slope of a velocity–time graph gives the instantaneous acceleration.
• The area under a velocity–time curve gives the displacement.
• The area under an acceleration–time curve gives the change in velocity.

Graphs connect position, velocity, and acceleration through slopes and areas.

Calculus Framework

• Velocity is the time derivative of position.
• Acceleration is the time derivative of velocity.
• These relationships invert through definite integrals: velocity is the integral of acceleration over time, and position is the integral of velocity over time.

When acceleration is not constant, the standard kinematic equations fail, and differentiation or integration must be used directly with the given functions.