The acceleration of a particle varies with time according to...

The acceleration of a particle varies with time according to the equation $a(t)=p t^{2}-q t^{3} .$ Initially, the velocity and position are zero. (a) What is the velocity as a function of time? (b) What is the position as a function of time?

Key Concepts

Kinematic Relationships

Kinematic relationships describe how acceleration, velocity, and displacement (position) of a particle are interrelated through derivatives and integrals. In these relationships, acceleration is the derivative of the velocity function with respect to time, and velocity is the derivative of the displacement with respect to time. Conversely, by integrating the acceleration function, one can determine the velocity function, and by integrating the velocity function, the displacement or position function is obtained.

Integration Techniques in Calculus

Integration in calculus is the process of finding a function whose derivative is a given function. In the context of kinematics, integrating the acceleration function yields the velocity function, and a subsequent integration of the velocity function yields the position function. The power rule is frequently used when integrating polynomial functions, where each term in the polynomial is integrated individually by increasing its exponent by one and then dividing by the new exponent.

Application of Initial Conditions

Initial conditions are crucial in solving problems involving integration in physics because the process of integration introduces arbitrary constants. These constants are then determined by applying the initial conditions of the problem, such as the initial velocity or initial position. This step ensures the derived functions accurately describe the motion of the particle from the specified starting point.

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