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Point kinematics online: velocity, acceleration and trajectory from x(t), y (t)

Enter the equations of motion x(t), y(t) and the time t₁. The calculator finds the trajectory equation, velocity, total, tangential and normal acceleration and the radius of curvature, shows the derivatives step by step and plots the trajectory with v and a vectors.

Write as on paper: 4sin(πt/6), 2t^2 + 3, e^(−2t), sqrt(t). Use pi for π and ^ for powers; the multiplication sign may be omitted. Functions: sin, cos, tan, cot, arcsin, arctan, ln, exp, sqrt.
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More examples

  • — parabola: x is linear in t
  • — parabola via cos 2u = 1 − 2sin²u
  • — parabola x(y): y is linear in t

Coordinate description of motion

Plane motion of a point is given by x = x(t), y = y(t). The classic first problem of theoretical mechanics: find the trajectory and all kinematic quantities at time t₁.

Solution steps

  1. Trajectory. Eliminate t. If one coordinate is linear in t, express t and substitute. With sin and cos of the same argument use sin²u + cos²u = 1 (ellipse or circle); with cos 2u and sin u use cos 2u = 1 − 2sin²u (parabola).
  2. Velocity. Components are first derivatives: vₓ = ẋ, v_y = ẏ; v = √(vₓ² + v_y²), tangent to the path.
  3. Acceleration. Second derivatives: aₓ = ẍ, a_y = ÿ; a = √(aₓ² + a_y²).
  4. Tangential acceleration aτ = dv/dt = (vₓaₓ + v_ya_y)/v; positive — speeding up, negative — slowing down.
  5. Normal acceleration an = √(a² − aτ²) = |vₓa_y − v_yaₓ|/v, pointing to the centre of curvature.
  6. Radius of curvature ρ = v²/an.

Example

x = 4sin(πt/6), y = −3cos(πt/6) + 4 (cm), t₁ = 1 s. From sin² + cos² = 1 the path is the ellipse x²/16 + (y − 4)²/9 = 1. Press «Example» to see all steps and the plot.

Check: aτ² + an² must equal a², and v must be tangent to the path.

FAQ

How do I find the trajectory equation from x(t) and y(t)?
Eliminate time: express t from one equation and substitute it into the other; for trigonometric motion use sin²u + cos²u = 1 or cos 2u = 1 − 2sin²u.
How are tangential and normal accelerations found?
aτ = (vₓaₓ + v_ya_y)/v, an = √(a² − aτ²). Velocity and acceleration components are the first and second time derivatives of the coordinates.
What is the radius of curvature?
The radius of the circle that best fits the path at the point: ρ = v²/an. On a straight segment an = 0 and ρ = ∞.
Is the motion accelerated or decelerated?
Look at the sign of the tangential acceleration (or of the dot product v·a): positive — accelerated, negative — decelerated.
Can I use it for coursework problems?
Yes: enter your equations and t₁. The calculator shows derivatives and substitutions — compare with your textbook and write up the solution yourself.