Final answer:
The velocity of the motorcycle is given by the function v(t) = (A/2)t² - (B/3)t³ + C, where A and B are constants. The position of the motorcycle is given by the function x(t) = (A/6)t³ - (B/12)t⁴ + Ct + D. The maximum velocity that the motorcycle attains is v_max = (A/4B).
Step-by-step explanation:
The velocity of the motorcycle can be found by integrating the acceleration function with respect to time. The integral of At - Bt² with respect to time is given by (A/2)t² - (B/3)t³. Therefore, the velocity as a function of time is represented by v(t) = (A/2)t² - (B/3)t³ + C, where C is the constant of integration.
To find the position as a function of time, we need to integrate the velocity function with respect to time. The integral of (A/2)t² - (B/3)t³ with respect to time is given by (A/6)t³ - (B/12)t⁴ + Ct + D, where D is the constant of integration. Therefore, the position as a function of time is represented by x(t) = (A/6)t³ - (B/12)t⁴ + Ct + D.
The maximum velocity that the motorcycle attains can be determined by finding the maximum value of the velocity function. The maximum occurs at the vertex of the parabolic function, which can be found using the vertex formula. The vertex formula for a quadratic function of the form y = ax² + bx + c is given by x = -b/2a. Plugging in the values for A and B, we get the maximum velocity as v_max = (A/4B).