Introduction to Derivatives

How do you find velocity and acceleration from position?

Quick answer

To find velocity from a position function s(t), take its first derivative with respect to time: v(t) = s'(t). To find acceleration, take the derivative of velocity, which is the second derivative of position: a(t) = v'(t) = s''(t). Setting velocity to zero identifies when an object stops or changes direction.

The topic explained

Derivatives of the position function

In physics and calculus, an object's location along a straight line at time t is modeled by a position function, written as s(t) or x(t). Because velocity measures the instantaneous rate at which position changes, velocity is the first derivative of position: v(t) = s'(t). A positive velocity means the object is moving forward or right, while a negative velocity means it is moving backward or left.

Acceleration measures the instantaneous rate at which velocity changes over time. Because of this, acceleration is the derivative of velocity, which makes it the second derivative of the position function: a(t) = v'(t) = s''(t). If position is measured in meters and time in seconds, velocity is in meters per second (m/s), and acceleration is in meters per second squared (m/s^2).

Finding rest points and turning points

An object is momentarily at rest whenever its instantaneous velocity equals zero. To find these times, set v(t) = 0 and solve for t. For example, if s(t) = t^3 - 6t^2 + 9t, then v(t) = 3t^2 - 12t + 9. Setting 3(t - 1)(t - 3) = 0 reveals that the object stops at t = 1 and t = 3.

Stopping is not the same as changing direction. An object only changes direction if the sign of v(t) flips from positive to negative or from negative to positive across that rest time. Test values on either side of each root or inspect the derivative's graph to verify whether a genuine reversal took place.

Deciding if an object is speeding up or slowing down

Speed is the magnitude of velocity, defined as |v(t)|. An object is speeding up whenever velocity and acceleration share the same sign. If both v(t) and a(t) are positive, the object is moving forward and being pushed forward. If both are negative, the object is moving backward and being pushed further backward.

An object is slowing down whenever velocity and acceleration have opposite signs. If v(t) is positive while a(t) is negative, the object is moving forward while being pulled back. To analyze this across an interval, create a sign chart comparing the signs of v(t) and a(t) simultaneously.

Finding maximum distance on an interval

To find the greatest distance from the starting point on a closed time interval [0, T], compare the absolute displacement |s(t) - s(0)| at all critical points where v(t) = 0 within the interval, as well as at the endpoint t = T. The largest of these values gives the maximum distance reached.

Free printable worksheet

Motion along a Line: Velocity and Acceleration Practice

Designed for 12th-grade calculus students. Differentiate each position function to find velocity and acceleration, then answer the questions about direction and speed.

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Motion along a Line: Velocity and Acceleration PracticeName:

Given s(t) = t^2 - 8t + 12 (in meters, for t >= 0), find v(t) and a(t).
Using s(t) = t^2 - 8t + 12, at what time is the body momentarily at rest?
For s(t) = t^3 - 3t^2, find all times t >= 0 when the object changes direction.
A particle moves with position s(t) = -t^2 + 6t. At t = 4, is the particle speeding up or slowing down?
Given s(t) = 2t^3 - 9t^2 + 12t on the interval [0, 3], find all intervals where the particle is moving to the left.
For s(t) = t^3 - 6t^2 on [0, 5], determine the greatest distance the body reaches from its starting point s(0) = 0.

Common questions

What is the difference between velocity and speed in calculus?

Velocity is a signed quantity representing both how fast an object moves and in what direction. Speed is the absolute value of velocity, meaning it only represents magnitude and is never negative.

Can an object have an acceleration of zero while in motion?

Yes. If an object moves at a constant velocity, its rate of change of velocity is zero. For example, if s(t) = 5t, velocity is a constant 5 m/s, so acceleration a(t) = 0 m/s^2.

Why does an object slow down when acceleration is negative?

An object only slows down when acceleration has the opposite sign of velocity. If velocity is positive and acceleration is negative, it slows down. However, if velocity is already negative, a negative acceleration actually speeds it up in the reverse direction.

How do you calculate total distance traveled versus displacement?

Displacement is simply s(end) - s(start). Total distance accounts for turns: split the time interval at every point where velocity equals zero, evaluate the absolute displacement across each subinterval, and add those values together.

Can Lessi explain step-by-step calculus homework problems?

Yes. Lessi is an AI teacher that can guide you through differentiation, sign charts, and optimization. The first 60 minutes of lessons with Lessi are free with an account, and the Ask Lessi box uses those same free minutes.

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