The material point moves in a straight line according to the law x (t) = 1 / 3t ^ 3-3t ^ 2-5t + 3

The material point moves in a straight line according to the law x (t) = 1 / 3t ^ 3-3t ^ 2-5t + 3, where x is the distance from the reference point in meters, t is the time in seconds measured from the moment the movement began. At what point in time (in seconds) was its speed equal to 2 m / s?

1. Let us calculate the first derivative of the function of the distance of a point on time, which expresses the dependence of the speed on time:

x (t) = 1/3 * t ^ 3 – 3t ^ 2 – 5t + 3;
v (t) = x ‘(t);
v (t) = (1/3 * t ^ 3 – 3t ^ 2 – 5t + 3) ‘;
v (t) = 1/3 * 3t ^ 2 – 3 * 2t – 5;
v (t) = t ^ 2 – 6t – 5.
2. Find the time moment for which v (t) = 2 m / s:

t ^ 2 – 6t – 5 = 2;
t ^ 2 – 6t – 7 = 0;
D / 4 = (b / 2) ^ 2 – ac = 3 ^ 2 + 7 = 16;
t = 3 ± √16 = 3 ± 4;
t1 = 3 – 4 = -1 (s);
t2 = 3 + 4 = 7 (s).
Answer. The point speed was 2 m / s at the time t = -1 s.