Impulse and Impulse-Momentum Theorem Physics Notes
Impulse and Impulse-Momentum Theorem:
A force which acts on a body for short interval of time is called impulsive force or impulse.
For Example Hitting, jumping, diving, catching etc. are all examples of impulsive forces or impulse.
An impulsive force does not remain constant, but changes first from zero to maximum and then from maximum to zero. Thus it is not possible to measure easily the value of impulsive force because it changes with time. In such cases, we measure the total effect of the force, called impulse hence, impulse is defined as the product of the average force and the time interval for which the force acts.
If \(\vec{F}\) is the value of force during impact at any time and \(\vec{p}\) is the momentum of the body at that time, then according to Newton’s second law of motion,
\(\vec{F}\) = \(\frac{d \vec{p}}{d t}\)
or \(\vec{F}\) dt = d\(\vec{p}\)
Suppose that the impact lasts for a small time t and during this time, the momentum of the body changes from \(\vec{p}_{1}\) to \(\vec{p}_{2}\) then integrating the above equation between the proper limits, we have:
It may be noted that F has not been taken out of the integration sign for the reason that varies with time and does not remain constant during impact. The integral ∫t0 F dt is measure of the impulse, when the force of impact acts on the body and from equation (1) we find that it is equal to total change in momentum of the body. Since impulse is equal to a scalar (time) times a vector (force) or equal to the change in momentum (vector), it is a vector quantity and it is denoted by I.
However if \(\vec{F}_{a v}\) is the average force (constant) during the impact, then
i. e., the change in momentum of an object equals the impulse applied to it. This statement is called
impulse-momentum theorem.
I = Δp
Dimensional Formula and Unit:
I = FΔt = [M1L1T-2] |T1]= [M1L1T-1]
So, the dimensional formula of impulse is the same as that of momentum.
The SI unit of impulse are (N-s) and kg m/s.
In C.G.S. system unit of impulse are dyne-sec and g cm/s.
Force Time Curve: In the real world, forces are often not constant. Forces may build up from zero over time and also may vary depending on many factors.
Finding out the overall effect of all these forces directly would be quite difficult. As we calculate impulse, we multiply force by time. This is equivalent to finding out the area under a force time curve. For variable force the shape of the force-time curve would be complicated but for a constant force we will get a simpler rectangle. In any case, the overall net impulse only matters to understand the motion of an object following an impulse.
In the figure, the graph of change in impulsive force with the time is shown:
The force-time curve and the area between the time axis can be divided in the form of many slabs. Suppose the value of force F is considered as constant along the change in time dt, then area of slab is given by F. dt.
The total effect of the force for time t1 to t2
= ∫t0 Fdt = sum of area of all slabs o
= graph of force-time and area covered between the time axis
∵ By the impulse-momentum theorem t
Impulse I = ∫t0 Fdt = p2 – p1 = change in momentum.
Thus force-time graph and the area covered with the time axis is equal to the total change in the momentum of the body.
Applications of the Concept of Impulse:
1. While catching a ball, a cricket player lowers his hands to save himself from getting hurt:
By lowering his hands, the cricket player increases the time interval in which the catch is completed. As the total change in momentum takes place in a large time interval, the time rate of change of momentum of the ball decreases. So, according to Newton’s second law of motion, lesser force acts on the hands of player and he saves himself from getting hurt.
2. Cars, buses, trucks, bogies of train etc. are provided with a spring system (shockers) to avoid severe jerks:
When they move over an uneven road, impulsive forces are exerted by the road. The function of shockers is to increase the time of impact. This would reduce the force/jerks experienced by the rider of the vehicle.
3. An athlete is advised to come to stop slowly: After finishing a fast race an athlete is advised to come to stop slowly, so that time of stop increases and hence force experienced by him decreases.
4. In a head-on collision: Between two vehicles change in linear momentum is equal to sum of the linear momentum of the two vehicles. As the time of impact is small, an extremely large force develops which causes damage to the vehicle.