A material point moves in a straight line in such a way that the speed changes according to the law V = 10-2t + 3t²

A material point moves in a straight line in such a way that the speed changes according to the law V = 10-2t + 3t² (time in seconds) Determine which way it will go in 5 seconds. Determine the acceleration of a point at a time of 5 seconds.

Data: the law of speed change: V (t) = 10 – 2t + 3t ^ 2; t5 – considered moment of time (t5 = 5 s).

1) The law of coordinate change (we find the antiderivative of the law of speed change): S (t) = ∫ (10 – 2t + 3t ^ 2) = 10t – t ^ 2 + t ^ 3 + C.

2) Path in 5 seconds: S = S5 – S0 = 10 * 5 – 52 + 53 + C – 10 * 0 – 0t ^ 2 – 03 – C = 50 – 25 + 125 = 150 m.

3) The law of acceleration change (we find the first-order derivative of the law of speed change): a (t) = (10 – 2t + 3t ^ 2) ‘= 0 – 2 * 1 + 3 * 2 * t = -2 + 6t.

4) Acceleration at time t = 5 s: a (5) = -2 + 6 * 5 = 28 m / s2.

Answer: In 5 seconds, the specified material daughter will travel 150 m, the acceleration will be 28 m / s2.