Calculus gives us a precise way to describe motion that changes from moment to moment. Position tells where an object is, velocity tells how position changes with time, and acceleration tells how velocity changes with time. These three quantities are linked by derivatives, so a graph of motion can reveal much more than just where an object has been.
This matters in physics, engineering, transportation, robotics, sports science, and any situation where motion must be predicted or controlled.
If s(t) is position, then velocity is the slope of the position graph, and acceleration is the slope of the velocity graph. The reverse process uses area: displacement comes from the area under a velocity time graph, and change in velocity comes from the area under an acceleration time graph. Positive and negative signs show direction, while the steepness of a graph shows how rapidly a quantity is changing.
Reading motion graphs well means connecting slope, area, sign, and units at the same time.
Key Facts
- Velocity is the derivative of position: v(t) = ds/dt.
- Acceleration is the derivative of velocity: a(t) = dv/dt = d2s/dt2.
- Displacement from time t1 to t2 is Δs = ∫ from t1 to t2 v(t) dt.
- Change in velocity from time t1 to t2 is Δv = ∫ from t1 to t2 a(t) dt.
- Average velocity is vavg = Δs/Δt, while instantaneous velocity is the slope of s(t) at one moment.
- Speed is the magnitude of velocity: speed = |v|, so it is never negative.
Vocabulary
- Position
- Position is an object's location relative to a chosen origin, usually measured in meters.
- Velocity
- Velocity is the rate of change of position with time, including both speed and direction.
- Acceleration
- Acceleration is the rate of change of velocity with time.
- Derivative
- A derivative gives the instantaneous rate of change of one quantity with respect to another.
- Displacement
- Displacement is the change in position from an initial point to a final point, including direction.
Common Mistakes to Avoid
- Confusing speed with velocity. Speed has no direction and is always nonnegative, while velocity can be positive, negative, or zero depending on direction.
- Reading height instead of slope on a position time graph. The value of the position graph gives location, but its slope gives velocity.
- Assuming negative acceleration always means slowing down. Negative acceleration means acceleration points in the negative direction, and an object speeds up if velocity is also negative.
- Ignoring units when moving between graphs. Position is measured in meters, velocity in meters per second, and acceleration in meters per second squared, so slopes and areas must match these units.
Practice Questions
- 1 A runner's position is s(t) = 3t2 + 2t, where s is in meters and t is in seconds. Find v(t), a(t), the velocity at t = 4 s, and the acceleration at t = 4 s.
- 2 A car has velocity v(t) = 12 - 2t in m/s from t = 0 s to t = 5 s. Find its acceleration, its displacement over the 5 s interval, and its speed at t = 5 s.
- 3 A position time graph is a curve that rises while getting flatter, then reaches a highest point, then slopes downward. Describe what happens to the object's velocity and explain whether the object changes direction.