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

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 in 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 (t) = 2 + 4 * t ^ 2 + t ^ 3.

t = 4 s.

an = 6 m / s ^ 2.

R -?

Let us write down the formula for determining the centripetal acceleration: an = V ^ 2 / R, where V is the speed of the body, R is the radius of the trajectory.

R = V ^ 2 / an.

The dependence of the speed of movement of the body is the derivative of the dependence of its movement on time: V (t) = S (t) “.

V (t) = S (t) “= (2 + 4 * t ^ 2 + t ^ 3.)” = 4 * 2 * t + 3 * t ^ 2 = 8 * t + 3 * t ^ 2.

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

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

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