The point is moving in a straight line with the speed v (t) = 8t ^ 3 + 9t ^ 2 + 3. Specify s (t) if s (1) = 2

1. The speed of a point is equal to the derivative of the coordinate function, therefore, the coordinate is determined by the integral of the speed:

v (t) = 8t ^ 3 + 9t ^ 2 + 3;
s (t) = ∫v (t) dt;
s (t) = ∫ (8t ^ 3 + 9t ^ 2 + 3) dt;
s (t) = 2t ^ 4 + 3t ^ 3 + 3t + s0, where s0 is a constant.
2. Let us find the value of s0, based on the value of the coordinate at time t = 1:

s (1) = 2;
s (t) = 2t ^ 4 + 3t ^ 3 + 3t + s0;
s (1) = 2 * 1 ^ 4 + 3 * 1 ^ 3 + 3 * 1 + s0 = 2 + 3 + 3 + s0 = 8 + s0;
8 + s0 = 2;
s0 = 2 – 8 = -6;
s (t) = 2t ^ 4 + 3t ^ 3 + 3t – 6.
Answer: s (t) = 2t ^ 4 + 3t ^ 3 + 3t – 6.