What is Impulse?

Impulse is defined as the force applied on an object (or body) for the regular interval of time. This in turn leads to the change in the linear momentum of the object and its direction. Impulse is used to quantify the effect of force acting over time to change the momentum of the object.

Impact of Impulse

Impulse may appear to be as force as we describe it as change in force. But it is important to note that when we divide the force in a large or small span of time, then this quantity comes into picture. More the intervals for which force act on a body or object, lesser will be its impact, and lesser it will hurt. Similarly, if the interval for which force acts is less, then it will have huge impact on the body and ultimately it will hurt more.

In order to understand it better, let us consider an example of a sofa. If one jumps on it instantaneously then the force that is exerted by that person on the sofa doesn’t vanishes immediately but will spread in an interval until the sofa gets fully compressed. So, one will not get injured compared to the situation when one jump from the height directly to the cemented surface. Where the force (or weight) of the person gets transferred to the floor instantaneously and in this case the impact the impulse is huge. So, one may get minor or even major injury. Therefore, it would be fair enough to say that the impulse is high or low depend on the period for which the force coming into picture is spread or divided.       

Consider another example of a truck carrying fragile or breakable material in thermocol boxes. And the truck stops instantaneously (in time t tending to zero) so that one of the boxes falls from the truck. Now the interval for which the thermacol box get compressed and undergoes Collision with the road completely is called impulse.

In this example, commercially impulse of the compressible materials are taken as an advantage to carry the fragile or breakable material such as glass pots, Television, mirrors, etc from one place to the other. Impulse prevents them from getting damaged on collision with some hard surfaces.

The SI unit of impulse is Newton second (N.s)

Derivation for expression of Impulse

Let’s consider a ball of constant mass “m” having an initial velocity “$v_i$”, now if an external force “F” is applied on it for the period “$\Delta t$”such that its final velocity changes to “$v_f$”. Now due to this external force, the acceleration “a” comes into picture. And which in turns into change in the momentum.

We know that according to Newton’s second law, the average force is the product of mass and acceleration:

$F=ma$ ---------(1)

Where, to calculate acceleration in terms of rate of change of initial velocity and final velocity, we use the equation:

$a=\frac{v_f-v_i}{\Delta t}$ -------------(2)

Now as we know that momentum is defined as:

$P = mv$

Consider two conditions:

Mass of the body changes and velocity is constant.

Mass of the body is constant and velocity changes.

In our case, mass of the ball is constant so this case comes under the category of second condition.

So, the final momentum in terms initial velocity of the ball is given by the equation:

$P=m{v}_i$ --------(3)

And the final momentum in terms of the final velocity of the ball is given by the equation:

$P=m{v}_f$ ------------(4)

Now, we insert (2) in (1) we get average Force equal to the rate of change of initial momentum and final momentum.

$F=\frac{m(v_f-v_i)}{\Delta t}$

Using (3) and (4)

$F=\frac{P_f-P_i}{\Delta t}$

Therefore, we can write the above expressions as:

$F=\frac{\Delta P}{\Delta t}$ --------------(5)

Equation (5) is stated as:

The net external average Force acting on the ball is changing the momentum of the ball in time interval $\Delta t$

Equation (5) can be written as:

$\Delta P=F*\Delta t$ ----- --------(6)

Where in physics the vector quantity termed as Impulse is given as: $F*\Delta t$ and hence impulse is defined as the change in momentum of the object or the momentum change of the object.

 Mathematically,

This is defined as the time integral of average force.

That is,

$I={\displaystyle {\int }_{-t}^{+t}Fdt}$ -----------(7)

The physical significance of the equation (1) is that if the average force F is applied on an object for the period t₁ to t₂ then the object experiences impulse. This impulse is a vector.

Also, $F=\frac{dp}{dt}$ --- -------(8)

Now, inserting (8) in (7), we get this vector quantity equal to momentum change.

$I={\displaystyle {\int }_{-t}^{+t}\frac{dp}{dt}}dt$

$I={\displaystyle {\int }_{-p_1}^{+p_2}dp}$

Therefore, we have derived the Formula Used to calculate vector that is, impulse, acting on the body when some external force acts on it for some period. In Physics, Impulse acting on a body is given by momentum change of the body.

$I=\Delta p$ -----------(9)

The above equation (9) is well known as the impulse- momentum theorem. Here, △p is the change in momentum of the object from period t₁ to t₂                            

According to this theorem, when an external force is applied to an body, not only there is change in velocity of the body from some initial velocity to some final velocity but also there is sufficient change from the initial momentum to final momentum of the body, hence an impulse is generated in it.

Use of impulse for Commercial Purposes:

The impulse is a very important concept used in our day-to-day life. For instance, while doing packaging of the breakable materials, this concept is used in hospitals, industries, mines, etc. The area under the curve of force V s time is plotted to find the safety limit of the fragile materials so that these sensitive materials can be delivered to their destination securely.  This is, by setting the limit of the fall we can estimate the change in the momentum and the average force that the object can bear without getting broken down.

Important Note:

 It is essential to remember that Impulse, force and momentum are vector quantities so one need to handle the given problem carefully keeping in mind the magnitude as well as direction of force and linear momentum to obtain the resultant impulse. Don’t forget to write the units with the final answer. While solving any numerical remember Newton’s Second law in mind to manipulate it to the required formula.

Formulas

The generalized formulas to calculate the Impulse and the related quantities of Physics are:

According to Newton’s second law: Force is equal to the product of mass and acceleration $F=ma$

Newton’s second law in terms of initial velocity and final velocity is equal to rate of change of momentum: $F=\frac{m(v_f-v_i)}{\Delta t}$

Force and momentum change are related as:

$F=\frac{dp}{dt}$

In terms of velocity, we have:

$I=m(v_f-v_i)$

The bodies undergoing collision experiences momentum change, in such a situation impulse is given as: $I=\Delta p$ Its unit of Newton’s seconds (NS).

Remember to calculate momentum in SI unit which in turns find change in momentum in SI unit, that is Newton’s seconds (NS).

Remember to analyze the physics of the collision of the two or more bodies to calculate the average force, initial and final velocity in similar unit.

Let’s take into account some practical situations where impulse comes into picture.

Practice Problems

Example 1:

Consider a cricket player playing on the field and while catching the ball during an almost sixer shot. He catches the ball at a height higher than himself and carried his hands downward. This will reduce the period of impact of the ball on the hand, which in turn will diminish the effect of force.  

Example 2:

Similarly, during long jump, the players practice on the heap of sand rather than on the hard-cemented floor. This is because of the yielding capability of the sand. Which diminishes the effect of force that one experiences when they fall on the sand and this increase in the interval of impact reduces the player from getting hurt on the sand.

Example 3:

Consider a ball of mass 5 Kg moving with the velocity 5m/s before it hits a wall. Calculate the impulse experienced by the ball.

Solution: 

$I=\Delta p$

$I=m(v_f-v_i)$

$\Delta p=m(v_f-v_i)$ = 5(0-5) = -25 Kg m/$s$

Note: The unit with the numeric result obtained.

Context and Applications

• Impulse plays important role to calculate the effects of elastic and inelastic collision on the body(s) under observation or the surface to which the object(s) impart force.

• If we suppose that the body under study has variable mass then impulse play very important background to calculate the motion of jets and the motion of rocket system. Since they impart huge force in almost no time and generate large impulse.  

• Note:  we have already discussed that there are two conditions where Impulse comes into picture, one is when mass is constant and velocity changes, which we have already discussed theoretically with examples. But the second case, that is, when mass of the body varies and its velocity is constant, such situation occurs in case of rocket science.

For instance, for the smooth motion of the rocket where the fuel (which is the mass in this case) continues to varies throughout its way until its final destination including its collision with dark matter in galaxy, the required impulse is analyzed throughout its motion. But this is out of scope at this level of study, so we haven’t discussed this condition in detail.

• Other countable applications of Impulse include:

1. Acoustic- optic modulator
2. Non-linear optics
3. Compton effect
4. wave particle duality in a wave collision, etc.

• Delivery of fragile products from one place to the other in the thermacol boxes or high stress bearing cardboard boxes to increase the period of impact and reduce the effect of force hence acceleration in the unfortunate situations of accidents where it may undergo exterior collision or interior collision with hard matter in its journey and mis-happening during transporting in trucks, buses etc.