# The body moves in a straight line with a speed v (t), and at time t0 the distance traveled is equal to s0.

The body moves in a straight line with a speed v (t), and at time t0 the distance traveled is equal to s0. Find the dependence of the path traveled by the body on time, if: v (t) = 3t ^ 2, t0 = 2s, s0 = 2 m;

Let us find the antiderivative of the velocity function v (t) = 3 * t²:
S (t) = 3 * t³ / 3 + C = t³ + C.

We calculate the constant C, substituting the known values of time t0 and displacement S0 into the resulting expression:
t0 = 2 s; S0 = 2 m.

S (t0) = t0³ + C;
2 = 2³ + C
C = 2 – 2³;
C = – 6.

Let us write down the dependence of the path traversed by the body on time, substituting the found constant C:
S (t) = t³ – 6.

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