# Impulse theorem

The impulse theorem shows that a force applied over a certain period of time on a body can generate a change in momentum.

The so-called **impulse theorem** shows that the impulse of a resultant force exerted on any object during a certain time interval is exactly equal to the change in the momentum of that object. Therefore, we have:

**I = ΔQ**

To understand this relationship between impulse and momentum, we can start from our daily lives. Daily experience shows us that the longer the time interval between the application of a force to an object, the greater the effect produced in relation to the speed of the body. The **impulse** (I) is the vector quantity that relates the resultant force (F _{R} ) and the time interval (Δt) of its application, being mathematically defined as the product of these two quantities.

I = F _{R.} Δt

It is possible to establish a relationship between the impulse and the momentum of a body to prove that the product of the resultant force by the time the force acts on any body generates variations in the momentum . For this, we will use Newton’s Second Law , in which the resultant force is given by the product of the mass of the body and its acceleration.

F _{R} = ma

Knowing that acceleration is the result of the ratio between the change in velocity and the change in time, we can rewrite Newton’s Second Law as:

F _{R} = m. __Δv__

Δt

F _{R }_{. }Δt = m. Δv

F _{R }_{. }Δt = m. (v – v _{0} )

F _{R }_{. }Δt = mv – mv _{0}

As the momentum (Q) is defined by the product of the body’s mass and its velocity, we have:

F _{R }_{. }Δt = _{FINAL} Q – _{INITIAL}

Q F _{R }_{. }Δt = ΔQ

Knowing that the product F _{R }_{. }Δt is the impulse, so we have that I = ΔQ.

The athlete, when hitting the tennis ball with the racket, gives it a boost

**Examples of Impulse application**

There are numerous situations in which the application of an impulse can be observed. We can cite the impulse acquired by the swimmer by jumping on a trampoline; the impulse offered to a soccer ball at the moment it is kicked; the catapult system that propels supersonic planes onto the decks of aircraft carriers; the initial speed acquired by bobsleigh athletes (sport portrayed in the image that opens the text), which depends on the impulse given at the time of the race, and so on.