A particle of mass m initially at rest, is acted upon by a...

A particle of mass $m$ initially at rest, is acted upon by a variable force $F$ for a brief interval of time $T$. It begins to move with velocity $u$ after the force stops acting. $F$ is as a function of time shown in the graph. The velocity o $[$ the particle is: (a) $u-\frac{\pi F_{e}^{2}}{2 m}$ (b) $u-\frac{\pi T^{2}}{8 m}$ (c) $u-\frac{\pi F_{n} T}{4 m}$ (d) $u-\frac{F_{n} T}{2 m}$

A particle of mass $m$ initially at rest, is acted upon by a variable force $F$ for a brief interval of time $T$. It begins to move with velocity $u$ after the force stops acting. $F$ is as a function of time shown in the graph. The velocity o $[$ the particle is:
(a) $u-\frac{\pi F_{e}^{2}}{2 m}$
(b) $u-\frac{\pi T^{2}}{8 m}$
(c) $u-\frac{\pi F_{n} T}{4 m}$
(d) $u-\frac{F_{n} T}{2 m}$

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Key Concepts

Impulse-Momentum Theorem

This theorem states that the total impulse applied to an object is equal to its change in momentum. In problems involving variable forces, integrating the force over the time interval gives the impulse, which, when divided by the mass, yields the change in velocity. It is a fundamental principle in dynamics, linking force and motion.

Impulse

Impulse refers to the product of a force and the time interval over which it acts. For variable forces, it is calculated as the integral of the force with respect to time. This concept is crucial for determining the change in momentum and, subsequently, the velocity of an object.

Force-Time Graph Interpretation

A force-time graph displays how a force changes with time. The area under this graph represents the impulse delivered to an object. Understanding how to interpret these graphs is essential for solving dynamics problems where the force is not constant, as it allows one to extract the integral of the force over time.

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