Motion in one dimension can be described by three linked functions: acceleration, velocity, and position. Calculus connects these functions through derivatives and antiderivatives. If you know acceleration, you can recover velocity by finding an antiderivative.
If you know velocity, you can recover position the same way, but you must also use initial conditions to fix the constants.
An antiderivative gives a whole family of possible functions because adding a constant does not change the derivative. In motion problems, those constants come from information such as starting velocity or starting position. For example, if a(t) = 6t and v(0) = 4, then v(t) = 3t^2 + 4.
Continuing with s(0) = 10 gives s(t) = t^3 + 4t + 10, which predicts the object's position at any time t.
Understanding Calculus: Antiderivatives and Motion
An integral in a motion problem has a physical meaning. It adds up many tiny changes over time. On an acceleration versus time graph, the signed area between the graph and the time axis gives the change in velocity over that interval.
Area above the axis increases velocity in the chosen positive direction. Area below the axis decreases it. The units confirm this idea.
Acceleration has units of metres per second per second. Multiplying by seconds leaves metres per second, which is a velocity unit. This unit check is a useful way to catch mistakes.
Initial conditions are not minor details. They describe the state of the object at one known time. Two cars can have exactly the same acceleration throughout a trip while having different velocities because they began differently.
They can then have different positions even if their velocity changes match. Starting position depends on the chosen reference point. A position of zero might mean a school gate, a finish line, or a sensor location.
The constant added after integration carries the units of the quantity being found. A velocity constant has velocity units. A position constant has distance units.
For common algebraic models, reversing differentiation follows a pattern. To find an antiderivative of a power of time, increase its exponent by one, then divide the coefficient by that new exponent. For instance, a term with coefficient six times time has an antiderivative with coefficient three times time squared.
A constant acceleration produces a velocity that changes linearly with time. It produces a position model containing a time squared term. The safest check is to differentiate the final position expression twice.
The first derivative should match the velocity expression. The second derivative should match the given acceleration.
Graphs make direction changes easier to see. A velocity graph crossing zero means the object is momentarily at rest. It may reverse direction at that point, though the surrounding velocity values must be checked.
Position can decrease when velocity is negative, even when the distance travelled keeps increasing. The integral of velocity gives displacement, which is the net change in position. Total distance requires adding the sizes of separate positive and negative movements.
Real motion data from phones, cars, or lab sensors is often noisy and may change in pieces. Students should identify the time interval, sign convention, units, and initial values before calculating anything. These choices determine whether a result describes a real trip correctly.
Key Facts
- v(t) = integral a(t) dt
- s(t) = integral v(t) dt
- a(t) = v'(t) = s''(t)
- v(t) = v0 + integral from 0 to t of a(u) du
- s(t) = s0 + integral from 0 to t of v(u) du
- If a(t) = 6t, v(0) = 4, and s(0) = 10, then v(t) = 3t^2 + 4 and s(t) = t^3 + 4t + 10
Vocabulary
- Antiderivative
- An antiderivative of a function f(t) is a function F(t) whose derivative is f(t).
- Acceleration
- Acceleration is the rate at which velocity changes with time.
- Velocity
- Velocity is the rate at which position changes with time, including direction.
- Position
- Position is an object's location along an axis relative to a chosen origin.
- Initial condition
- An initial condition is a known value, such as s(0) or v(0), used to determine the constant in an antiderivative.
Common Mistakes to Avoid
- Forgetting the constant of integration, which is wrong because many functions can have the same derivative.
- Using only a(t) to find s(t) in one step without two integrations, which is wrong because acceleration must first be integrated to velocity and then velocity to position.
- Ignoring initial conditions, which is wrong because the starting velocity and starting position determine the specific motion function.
- Mixing up signs for direction, which is wrong because positive and negative velocity or acceleration represent opposite directions along the chosen axis.
Practice Questions
- 1 A cart has acceleration a(t) = 8t m/s^2 and initial velocity v(0) = 3 m/s. Find v(t) and v(2).
- 2 A runner has velocity v(t) = 5t^2 + 2 m/s and initial position s(0) = 4 m. Find s(t) and s(3).
- 3 Two objects have the same acceleration function but different initial velocities. Explain how their velocity functions and position functions compare over time.