How to find the radius of curvature of a trajectory

How to find the radius of curvature of a trajectory

By consideration of motion of bodies a number of the characterizing sizes, for example tangential and normal (centripetal) acceleration, speed and also curvature of a trajectory is used. Curvature radius – the geometrical concept designating the radius of a circle of R on which the body moves. This parameter can be found on the corresponding formulas by means of the set trajectory of the movement.


1. Most often tasks on determination of radius of curvature of a trajectory of flight of the thrown body in the set period meet. The trajectory of the movement in this case is described by the equations on coordinate axes: x = f(t), y = f(t) where t is time at the time of which it is required to find radius. Its calculation will be based on application of a formula an = V²/R. Here the radius of R comes to light from the relation of normal acceleration of an and instantaneous velocity of the V movement of a body. Having learned these sizes, component R can find easily required.

2. Calculate body speed projections on axes (OX, OY). The mathematical meaning of speed is the first derivative of the equation of the movement. Therefore they easily are capture derivative of the set equations: Vx = x', Vy = y'. By consideration of geometrical display of these projections in a coordinate system it is visible that they are legs of a rectangular triangle. And a hypotenuse in it – required instantaneous velocity. Proceeding from it, calculate the size of instantaneous velocity V in Pythagorean theorems: V = √ (Vx² + Vy²). Substituting the known value of time in expression, find a numerical indicator of V.

3. It is also easy to define the module of normal acceleration, having considered other rectangular triangle formed by the module of full acceleration and and tangent acceleration of a body of the academician. And here normal acceleration is a leg and is calculated so: an = √ (and² - joint stock company²). For finding of tangent acceleration differentiate the equation of instantaneous velocity of the movement on time: joint stock company =. Full acceleration calculate |dV/dt| on its projections to axes, similar to finding of instantaneous velocity. Only for this purpose take derivatives of the second order from the set equations of the movement: ah = x'', ay = y''. Module of acceleration and = √ (akh2 + ay2). Substituting all found sizes, define numerical value of normal acceleration an = √ (and² - joint stock company²).

4. Express from a formula an = V²/R a required variable of radius of curvature of a trajectory: R = V \an. Substitute numerical values of speed and acceleration, calculate radius.

Author: «MirrorInfo» Dream Team