Work Sign from a Velocity-Time Graph

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Work and Energy Beginner work-energy

Source: High school physics (Chinese)

Problem Sets:

work-energy 1018

Problem

Under the action of a single variable force, an object undergoes linear motion. Its velocity-time graph is shown in the figure.

Determine the sign (positive, negative, or zero) of the work done by this force during the time intervals OB, BC, CD, and DE.
P0450-problem-1

P0450-problem-1

OB: Positive (+) BC: Zero (0) CD: Negative (-) DE: Positive (+)

According to the work-energy theorem, the net work done on an object equals its change in kinetic energy: $W_{net} = \Delta E_k = \frac{1}{2}m(v_f^2 - v_i^2)$. Since a single force acts on the object, it is the net force. The sign of the work depends on whether the speed increases, decreases, or remains constant.

Interval OB: The velocity increases from $v_O=0$ to $v_B>0$. The speed increases, so $\Delta E_k > 0$. The work done is positive. Interval BC: The velocity is constant ($v_C = v_B$). The speed does not change, so $\Delta E_k = 0$. The work done is zero. Interval CD: The velocity decreases from $v_C>0$ to $v_D=0$. The speed decreases, so $\Delta E_k < 0$. The work done is negative. Interval DE: The velocity changes from $v_D=0$ to a negative value $v_E$. The speed increases from 0 to $|v_E|$, so $\Delta E_k > 0$. The work done is positive.