The law of motion of a point along a curve is expressed by the equation S = 2 + 4t ^ 2 + t ^ 3. Find the radius of curvature

The law of motion of a point along a curve is expressed by the equation S = 2 + 4t ^ 2 + t ^ 3. Find the radius of curvature R of the trajectory at the place where this point will be located at the time t = 4 s, if the normal acceleration at this time is equal to an = 6 m / s.

S = 2 + 4 * t ^ 2 + t ^ 3.

t = 4 s.

an = 6 m / s ^ 2.

R -?

Normal or centripetal acceleration an when moving around a circle is determined by the formula: an = V ^ 2 / R, where V is the speed of movement at a given time, R is the radius of the trajectory of movement.

R = V ^ 2 / an.

The dependence of the speed of movement V (t) is the first derivative of the distance traveled S “(t): V (t) = S” (t).

V (t) = (2 + 4 * t ^ 2 + t ^ 3) “= 4 * 2 * t + 3 * t ^ 2.

V (4 s) = 8 * 4 + 3 * (4 s) ^ 2 = 80 m / s.

R = (80 m / s) ^ 2/6 m / s ^ 2 = 1066.6 m.

Answer: the radius of curvature of the trajectory is R = 1066.6 m.