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:

    “All science requires mathematics. The knowledge of mathematical things is almost innate in us.... This is the easiest of sciences, a fact which is obvious in that no one’s brain rejects it; for laymen and people who are utterly illiterate know how to count and reckon.”
    —Roger Bacon (c. 1214–c. 1294)

    “It was a maxim with Mr. Brass that the habit of paying compliments kept a man’s tongue oiled without any expense; and that, as that useful member ought never to grow rusty or creak in turning on its hinges in the case of a practitioner of the law, in whom it should be always glib and easy, he lost few opportunities of improving himself by the utterance of handsome speeches and eulogistic expressions”
    —Charles Dickens (1812–1870)

    “If thinking is like perceiving, it must be either a process in which the soul is acted upon by what is capable of being thought, or a process different from but analogous to that. The thinking part of the soul must therefore be, while impassable, capable of receiving the form of an object; that is, must be potentially identical in character with its object without being the object. Mind must be related to what is thinkable, as sense is to what is sensible.”
    —Aristotle (384–322 B.C.)

    “A constant smirk upon the face, and a whiffling activity of the body, are strong indications of futility.”
    —Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)

    “Reduced to a miserable mass level, the level of a Hitler, German Romanticism broke out into hysterical barbarism.”
    —Thomas Mann (1875–1955)