AP Syllabus focus:
‘Understand that the velocity function is the derivative of the position function with respect to time and that velocity describes both the speed and direction of motion.’
Velocity connects how an object’s position changes over time to instantaneous motion. Understanding velocity as the derivative allows precise descriptions of both speed and directional behavior in straight-line motion.
Velocity as the Derivative of Position
In straight-line motion, the key idea is that velocity represents how fast and in what direction an object’s position changes at a given moment. When position is expressed as a function of time, velocity emerges directly from differentiation, linking calculus to physical motion.
Position as a Function of Time
We describe the motion of an object along a line by a position function, typically written as or , where represents time. The value of the position function indicates the exact location of the moving object at any instant.
Position Function: A function that assigns the location of an object on a line to each moment in time.
Because motion rarely proceeds at a constant rate, calculus provides tools to determine how fast the position changes at any specific instant.
Velocity as the Instantaneous Rate of Change of Position
Velocity in straight-line motion is defined as the instantaneous rate of change of the position function with respect to time. This connects directly to the derivative, since derivatives measure how one quantity changes relative to another at a precise moment.
Velocity: The instantaneous rate of change of position with respect to time; it includes both speed and direction of motion.
Velocity includes direction, distinguishing it from speed. Positive velocity indicates movement in the positive direction along the line, while negative velocity indicates movement backward or in the opposite direction.
A full mathematical representation helps reinforce this relationship.
= Velocity at time (units of position per unit time)
= Derivative of the position function with respect to time
This expression highlights that velocity is not simply a number describing how fast something moves, but a derivative revealing the object’s directional and instantaneous behavior.
A single sentence is placed here to separate the equation block from the next equation, as required.
= Position (units of length)
= Time (units of time)
Interpreting Velocity in Straight-Line Motion
Understanding velocity as the derivative enables students to describe motion with precision. When interpreting velocity:
A positive velocity signals movement in the positive direction.
A negative velocity signals movement in the opposite direction.
A velocity of zero indicates a momentary stop or a change in direction.
These interpretations allow the derivative to serve as a conceptual bridge between algebraic expressions and physical movement.
A collection of position–time graphs showing how different slopes represent positive, negative, or zero velocity. Each line or curve illustrates how the slope corresponds to an object’s instantaneous velocity. Some curves also imply changing velocity, adding minor detail beyond this subsubtopic. Source.
Distinguishing Speed from Velocity
Although the terms “speed” and “velocity” are related, they are not interchangeable. Speed is the magnitude of velocity and therefore is always nonnegative. Velocity, on the other hand, communicates both magnitude and direction.
Speed: The magnitude of velocity, representing how fast an object moves regardless of direction.
Because velocity carries sign information, analyzing its behavior helps determine when an object is reversing direction or maintaining a specific orientation along its path.
Graphical Views of Position and Velocity
The relationship between position and velocity becomes especially clear when examining graphs:
The slope of the position graph at any point represents the instantaneous velocity.
A steeper slope corresponds to a higher speed.
A slope of zero corresponds to a moment when the object is momentarily at rest.
Negative slopes indicate motion in the negative direction.
Graphical interpretation helps reinforce that velocity is fundamentally a derivative concept, rooted in the geometry of tangent lines.
A position–time graph illustrating the difference between average and instantaneous velocity. The secant line indicates average velocity over an interval, while the tangent line indicates instantaneous velocity at a given moment. The labeling of average velocity adds minor extra detail but strongly supports understanding slope-based interpretation. Source.
How Differentiation Reveals Motion
Differentiation transforms a position function into a velocity function by capturing the instantaneous change in location. This process allows us to:
Determine the object’s direction of motion at any given time.
Identify when the object speeds up or slows down (later examined through acceleration).
Understand detailed behavior that average rate of change cannot reveal.
Using Units to Interpret Velocity
Correct interpretation in context requires tracking units carefully. Since velocity is the derivative of position with respect to time, its units always follow the form:
(units of position) ÷ (units of time)
Examples include meters per second, feet per second, or miles per hour.
Precise unit interpretation ensures that descriptions of motion are consistent with the problem’s context.
Key Takeaways for AP Calculus AB
Velocity as the derivative of a position function provides a foundational connection between calculus and real-world motion. Recognizing that velocity conveys both speed and direction allows accurate interpretation of physical situations. Differentiating the position function remains the central tool for understanding instantaneous motion in straight-line contexts.
FAQ
Instantaneous velocity is a mathematical measure, not a direct observation. It captures how the position would change if the motion continued with the exact trend occurring at that moment.
It relies on imagining the position curve extremely close to the point of interest and treating it as linear. This is why differentiation is essential: it calculates the slope of that locally linear behaviour.
Velocity incorporates direction, so a negative velocity indicates movement opposite to the chosen orientation of the axis.
Speed, by contrast, is the magnitude of the velocity and is always positive. A particle with velocity −5 m/s and one with +5 m/s have the same speed but move in different directions.
Yes. The curve may not appear visually flat on a large scale, but at a sufficiently small scale, the tangent at a point can be horizontal.
A zero derivative occurs when the local slope is zero, even if the overall graph looks curved. This often happens at turning points where the motion reverses direction.
A steeper slope indicates quicker change in position, meaning greater instantaneous velocity.
Key relationships include:
• Very steep positive slopes: fast motion in the positive direction.
• Very steep negative slopes: fast motion in the negative direction.
• Gentle slopes: slower motion regardless of direction.
Velocity reveals direction and rate of change of position, but some motion details require additional derivatives or context.
Velocity alone cannot determine:
• Whether the particle is speeding up or slowing down.
• Whether the path includes turning points beyond where velocity is zero.
• How the motion behaves over an interval without examining the position function or acceleration.
Practice Questions
Question 1 (1–3 marks)
A particle moves along a straight line so that its position at time t seconds is given by x(t) = 4t − t² metres.
(a) Find the velocity of the particle at t = 3.
Question 1
(a)
• Differentiate x(t) = 4t − t² to obtain the velocity function v(t) = 4 − 2t. (1 mark)
• Substitute t = 3 to obtain v(3) = 4 − 6 = −2 m/s. (1 mark)
• Correct units and interpretation of negative sign (e.g., moving in the negative direction). (1 mark)
Question 2 (4–6 marks)
The position of a particle moving along a line is given by s(t) = t³ − 6t² + 9t metres for t ≥ 0.
(a) Find the velocity function.
(b) Determine the time at which the particle is momentarily at rest.
(c) State whether the particle is moving in the positive or negative direction immediately after that time, giving a reason with reference to velocity.
Question 2
(a)
• Differentiate s(t) = t³ − 6t² + 9t to obtain v(t) = 3t² − 12t + 9. (1 mark)
(b)
• Set the velocity equal to zero: 3t² − 12t + 9 = 0. (1 mark)
• Solve to obtain t = 1 or t = 3. (1 mark)
(c)
• Correct reasoning using a value of velocity just after t = 1 or t = 3, or using sign analysis of the velocity function. (1 mark)
• Correct statement of direction (positive or negative) based on the sign of v(t). (1 mark)
