Impulse (physics) - Mathematical Derivation in The Case of An Object of Constant Mass

Mathematical Derivation in The Case of An Object of Constant Mass

Impulse J produced from time t1 to t2 is defined to be

where F is the force applied from t1 to t2.

From Newton's second law, force is related to momentum p by

Therefore

\begin{align} \mathbf{J} &= \int_{t_1}^{t_2} \frac{d\mathbf{p}}{dt}\, dt \\ &= \int_{p_1}^{p_2} d\mathbf{p} \\ &= \Delta \mathbf{p}, \end{align}

where Δp is the change in momentum from time t1 to t2. This is often called the impulse-momentum theorem.

As a result, an impulse may also be regarded as the change in momentum of an object to which a force is applied. The impulse may be expressed in a simpler form when both the force and the mass are constant:

It is often the case that not just one but both of these two quantities vary.

In the technical sense, impulse is a physical quantity, not an event or force. The term "impulse" is also used to refer to a fast-acting force. This type of impulse is often idealized so that the change in momentum produced by the force happens with no change in time. This sort of change is a step change, and is not physically possible. This is a useful model for computing the effects of ideal collisions (such as in game physics engines).

Impulse has the same units (in the International System of Units, kg·m/s = N·s) and dimensions (M L T−1) as momentum.

Impulse can be calculated using the equation

where

F is the constant total net force applied,
t is the time interval over which the force is applied,
m is the constant mass of the object,
v1 is the final velocity of the object at the end of the time interval, and
v0 is the initial velocity of the object when the time interval begins.

Read more about this topic:  Impulse (physics)

Famous quotes containing the words mathematical, case, object, constant and/or mass:

    “The most distinct and beautiful statement of any truth must take at last the mathematical form.”
    —Henry David Thoreau (1817–1862)

    “The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
    —Samuel Taylor Coleridge (1772–1834)

    “It is the nature of our desires to be boundless, and many live only to gratify them. But for this purpose the first object is, not so much to establish an equality of fortune, as to prevent those who are of a good disposition from desiring more than their own, and those who are of a bad one from being able to acquire it; and this may be done if they are kept in an inferior station, and not exposed to injustice.”
    —Aristotle (384–322 B.C.)

    “Love not me for comely grace,
    For my pleasing eye or face,
    Nor for any outward part:
    No, nor for a constant heart!”
    —Unknown. Love Not Me for Comely Grace (l. 1–4)

    “The mass believes that it has the right to impose and to give force of law to notions born in the café.”
    —José Ortega Y Gasset (1883–1955)