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

“An accurate charting of the American woman’s progress through history might look more like a corkscrew tilted slightly to one side, its loops inching closer to the line of freedom with the passage of time—but like a mathematical curve approaching infinity, never touching its goal. . . . Each time, the spiral turns her back just short of the finish line.”
—Susan Faludi (20th century)

“Pardon me, you are not engaged to any one. When you do become engaged to some one, I, or your father, should his health permit him, will inform you of the fact. An engagement should come on a young girl as a surprise, pleasant or unpleasant, as the case may be. It is hardly a matter that she could be allowed to arrange for herself.”
—Oscar Wilde (1854–1900)

“The charm of fame is so great that we like every object to which it is attached, even death.”
—Blaise Pascal (1623–1662)

“Henry David Thoreau, who never earned much of a living or sustained a relationship with any woman that wasn’t brotherly—who lived mostly under his parents’ roof ... who advocated one day’s work and six days “off” as the weekly round and was considered a bit of a fool in his hometown ... is probably the American writer who tells us best how to live comfortably with our most constant companion, ourselves.”
—Edward Hoagland (b. 1932)

“Commercial jazz, soap opera, pulp fiction, comic strips, the movies set the images, mannerisms, standards, and aims of the urban masses. In one way or another, everyone is equal before these cultural machines; like technology itself, the mass media are nearly universal in their incidence and appeal. They are a kind of common denominator, a kind of scheme for pre-scheduled, mass emotions.”
—C. Wright Mills (1916–62)