Motion on a line with calculus – position, velocity, acceleration and distance
MathematicsCalculusAges 17–18
Loading…
Sign in to playChoose a position s(t), velocity v(t) or acceleration a(t) function (polynomial, trigonometric or exponential, including variable acceleration) and watch a particle move along a line with linked s–t, v–t and a–t graphs. The graphs are connected by differentiation and integration: shaded areas under the v–t graph separate displacement ∫v dt from distance ∫|v| dt, and the sim marks when the particle is at rest, turns around, speeds up or slows down.
Lesson: Kinematics with calculus: velocity and acceleration as derivatives, displacement and distance as integrals
What it shows
A particle moves along a straight line. Velocity is the derivative of position and acceleration is the derivative of velocity, so v = ds/dt and a = dv/dt. Going the other way, s(t) = s(0) + ∫v dt and v(t) = v(0) + ∫a dt, which is why initial conditions are needed when v or a is given. The signed area under the v–t graph is the displacement, while the area under |v| is the total distance. The particle turns around where v changes sign, speeds up when v and a share a sign and slows down when they differ.
How to use
Under Given, choose Position s(t), Velocity v(t) or Acceleration a(t), pick a Function type and set p, q, r and c. Enter s(0) and v(0) when they are needed. Press Play or drag across the graphs to move the time cursor, and compare each tangent slope with the graph below it. Read the displacement and distance in the panel, and try an Example such as the oscillation or the vertical throw.
Parameters you can change
- Given function Position s(t), Velocity v(t), Acceleration a(t)
- Function type Polynomial p·t³ + q·t² + r·t + c, Trigonometric p·sin(q·t) + r·cos(q·t) + c, Exponential p·e^(q·t) + r·t + c
- Coefficient p -20–20
- Coefficient q -10–10
- Coefficient r -20–20
- Coefficient c -20–20
- Initial position s(0) -20–20 m
- Initial velocity v(0) -20–20 m/s
- Time interval T 1–20 s
Questions to explore
- For s = t³ − 6t² + 9t, when does the particle turn around and what distance does it travel in 5 s?
- Why can the displacement be smaller than the distance travelled, and when are they equal?
- With a = 6 − 2t, how do the v–t graph and the turning time change if v(0) increases?
