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:

    “As we speak of poetical beauty, so ought we to speak of mathematical beauty and medical beauty. But we do not do so; and that reason is that we know well what is the object of mathematics, and that it consists in proofs, and what is the object of medicine, and that it consists in healing. But we do not know in what grace consists, which is the object of poetry.”
    —Blaise Pascal (1623–1662)

    “In all unmerciful actions, the worst of men pay this compliment at least to humanity, as to endeavour to wear as much of the appearance of it, as the case will well let them.”
    —Laurence Sterne (1713–1768)

    “The exercise of letters is sometimes linked to the ambition to contruct an absolute book, a book of books that includes the others like a Platonic archetype, an object whose virtues are not diminished by the passage of time.”
    —Jorge Luis Borges (1899–1986)

    “In verse one can take any damn constant one likes, one can alliterate, or assone, or rhyme, or quant, or smack, only one MUST leave the other elements irregular.”
    —Ezra Pound (1885–1972)

    “After the planet becomes theirs, many millions of years will have to pass before a beetle particularly loved by God, at the end of its calculations will find written on a sheet of paper in letters of fire that energy is equal to the mass multiplied by the square of the velocity of light. The new kings of the world will live tranquilly for a long time, confining themselves to devouring each other and being parasites among each other on a cottage industry scale.”
    —Primo Levi (1919–1987)