Up Linear Momentum

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Impulse of a constant Force

In order to introduce the definition of impulse, at the beginning of this unit, the net force and the acceleration are regarded as constant. The definition of impulse is developed while considering as starting point Newton's second law of motion, , where is the net external force acting on the object and the mass, m, and acceleration, ,of the object, along with the definition of acceleration, where is the change in velocity and is the time taken. In this case, because the acceleration is constant; thus, equal to its average. Combining these two expressions the following relation is obtained,

 

The last expression is the foundation for the definition of Impulse, . From this definition it is concluded that in order to know the effect of a net force acting on an object, it is not only necessary to precise the value of the force but it is also necessary to recognize the time that this force acts on the object. Further understanding of this definition is obtained studying the graphical representation of the impulse from the graph of the force versus the time.

In the graph at the left, the area under the curve of the force versus the time correspond to the impulse, . This definition can be extended to situations where the acceleration and/or force are variant. Thus,

Impulse is the area under the curve of the force versus the time.

 

The units of impulse are .

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Examples:  

 

1)

J

Impulse

 Q 1

 

 

Calculate the impulse associated to the force represented on the graph at the left.

 

 

a)

360 kg m/s

 

b)

240 kg m/s

 

c)

80 kg m/s

N

d)

380 kg m/s

 

e)

None of the above.

 

2)

J

Impulse

 Q 2

 

 

Calculate the impulse associated to the force represented on the graph at the left.

 

 

a)

420 kg m/s

  N

b)

245 kg m/s

 

c)

105 kg m/s

 

d)

140 kg m/s

 

e)

None of the above.

 

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Impulse of a variant Force

In most cases, the forces applied over objects are not constant or they do not have a simple variation with the time such like the case of the force applied by a golf club  to a golf ball. A variant force can also be the force applied by a soccer player when kicks the ball, a kicker of a football team when kicks the football, a hitter hitting a baseball with a bat, or a tennis player hitting the ball. In all these cases, the force can vary dramatically over a very short period of time. The following graphical representation illustrate the form in which the force can vary, as a function of the time, in these cases.

The graph on the left shows the time dependence of the force for one situation such as one of the mentioned above. The time interval where the force is different than zero, , can be very short, as short as 0.1 second or 0.01 second. However, the maximum value of the force can be very large. As mentioned above, the impulse is the area under the curve. Certainly, calculations made out with this functional dependence can be very difficult and, at the same time, unnecessary. For such short time periods, the exact detail of the force becomes immaterial (except when that is precisely what it is being study) and the following approach provides with most of the needed information. With this objective, an average force is defined by selecting a constant force acting over the same interval time with the condition that the area of the rectangle formed by the constant force has exactly the same area as the original variant force, , this force is called the average force, . Based on this definition, the impulse is .

The applet on the left shows a sequence of drawing illustrating the processes of matching the area of the actual peak representing the impulse to the area of the above defined average force.

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by Luis F. Sez, Ph. D.    Comments and Suggestions: LSaez@dallaswinwin.com