When acceleration be function of velocity as a=f(v). Then:...

When acceleration be function of velocity as $a=f(v)$. Then:
(a) the displacement $(x)=: \int \frac{\partial d v}{f(v)}$
(b) the acceleration may be constant
(c) the slope of acceleration versus velocity graph may be constant
(d) (a) and (c) are correct

Key Concepts

Differential Equations in Kinematics

In kinematics, acceleration is often expressed as a function of velocity, represented by the differential equation a = dv/dt = f(v). This relationship forms the basis for analyzing motion by linking the rates of change of velocity and displacement, and it provides a framework for solving problems in dynamics using calculus.

Separation of Variables and Integration

When acceleration is given as a function of velocity, the method of separation of variables can be used to rewrite the differential equation in an integrable form. By expressing dt in terms of dv (dt = dv/f(v)), and then writing the displacement as the integral of velocity times dt, one obtains an integral expression that connects displacement, velocity, and the functional form of acceleration.

Special Cases: Constant and Linear Acceleration

Certain choices of the function f(v) lead to well?known kinematic scenarios. For example, if f(v) is a constant, the acceleration is constant, which is a common situation in introductory mechanics. Alternatively, if f(v) is a linear function of velocity, the graph of acceleration versus velocity will have a constant slope, reflecting a steady rate of change of acceleration with respect to velocity.

Graphical Interpretation of Functional Relationships

The representation of acceleration as a function of velocity is not only useful in analysis but can also be visualized graphically. In such a graph, the slope (i.e., the derivative f’(v)) provides information about how acceleration changes with velocity. A constant slope indicates that this change is uniform, which corresponds to a linear functional dependence.

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