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.

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