A body is moved along a straight line by a machine...

A body is moved along a straight line by a machine delivering a constant power. The distance moved by the body in time $t$ is proportional to
(A) $t^{3 / 4}$
(B) $t^{3 / 2}$
(C) $t^{1 / 4}$
(D) $t^{1 / 2}$

Key Concepts

Constant Power

Constant power refers to a situation where the machine delivers energy at a fixed rate, meaning that the work done on an object increases linearly with time. This idea is central because it sets up the relationship between the energy transferred to the object and its kinetic energy, which in turn affects its velocity and displacement.

Work-Energy Principle

The work-energy principle states that the work done on an object is equal to the change in its kinetic energy. In the context of constant power, the rate at which work is done (power) is directly tied to how the kinetic energy—and consequently, the velocity—of the object increases over time.

Integration in Kinematics

When dealing with non-constant acceleration, such as that resulting from constant power, it is necessary to use calculus to relate kinematic quantities. Specifically, one must integrate the relation between the rate of work (or energy transfer) and acceleration to determine how velocity and displacement depend on time. This integration reveals the power-law relationship between distance and time.

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