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Motion graphs from calculus
Intermediate
Slopes and areas are derivatives and integrals of the motion graphs.
The three graphs are linked: x–t slope gives v, v–t slope gives a, and areas go the other way. At a peak of x(t), v = 0; at a peak of v(t), a = 0.
Given x = t³ − 6t², v = 3t² − 12t is zero at t = 4 s, where x is at a minimum.
Zero slope means zero of the next quantity, not zero of the quantity itself.
In more detail
Calculus links the motion graphs: velocity is the derivative of position, acceleration is the derivative of velocity, and displacement is the integral of velocity. These links let you handle curved or changing motion.
Real-world uses
- Physics simulators compute position and velocity by integrating acceleration step by step.
- Sports scientists differentiate measured position to get velocity and acceleration.
- Data loggers record a sensor's position, and software differentiates it to show velocity and acceleration graphs automatically.
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Frequently asked questions
Why is calculus needed in kinematics?
Because motion is described by rates of change, and calculus is the mathematics of rates.
What does a zero velocity mean in calculus terms?
The derivative of position is zero, so the position is at a maximum, minimum or turning point.
Why do graphs of x, v and a look related?
Each is the derivative of the one before it, so the shape of one graph determines the others.
Related topics in Kinematics
Stopping distance · Reaction time and braking · Non-uniform acceleration · Terminal velocity · Two-dimensional motion · Projectile motion