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Unformatted text preview: J. Peraire, S. Widnall 16.07 Dynamics Fall 2009 Version 2.0 Lecture L9- Linear Impulse and Momentum. Collisions In this lecture, we will consider the equations that result from integrating Newtons second law, F = m a , in time. This will lead to the principle of linear impulse and momentum. This principle is very useful when solving problems in which we are interested in determining the global effect of a force acting on a particle over a time interval. Linear Momentum We consider the curvilinear motion of a particle of mass, m , under the inuence of a force F . Assuming that the mass does not change, we have from Newtons second law, d v d F = m a = m = ( m v ) . dt dt The case where the mass of the particle changes with time (e.g. a rocket) will be considered later on in this course. The linear momentum vector, L , is defined as L = m v . Thus, an alternative form of Newtons second law is F = L , (1) which states that the total force acting on a particle is equal to the time rate of change of its linear momentum. Principle of Linear Impulse and Momentum Imagine now that the force considered acts on the particle between time t 1 and time t 2 . Equation (1) can then be integrated in time to obtain t 2 t 1 F dt = t 2 t 1 L dt = L 2 L 1 = L . (2) Here, L 1 = L ( t 1 ) and L 2 = L ( t 2 ). The term t 2 I = t 1 F dt = L = ( m v ) 2 ( m v ) 1 , is called the linear impulse . Thus, the linear impulse on a particle is equal to the linear momentum change L . In many applications, the focus is on an impulse modeled as a large force acting over a small time. But 1 in fact, this restriction is unnecessary. All that is required is to be able to perform the integral t 2 F dt . If t 1 the force is a constant F, then L = t 2 F dt = F ( t 2 t 1 ). If the force is given as a function of time, then t 1 L = t t 1 2 F ( t ) dt Note Units of Impulse and Momentum It is obvious that linear impulse and momentum have the same units. In the SI system they are N s or kg m/s, whereas in the English system they are lb s, or slug ft/s. Example (MK) Average Drag Force The pilot of a 90 , 000-lb airplane which is originally ying horizontally at a speed of 400 mph cuts off all engine power and enters a glide path as shown where = 5 o . After 120 s, the airspeed of the plane is 360 mph. We want to calculate the magnitude of the time-averaged drag force. Aligning the x-axis with the ight path, we can write the x component of equation (2) as follows 120 ( W sin D ) dt = L x (120) L x (0) . The time-averaged value of the drag force, D , is 120 1 D = D dt . 120 Therefore, ( W sin D )120 = m ( v x (120) v x (0)) ....
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This note was uploaded on 11/06/2011 for the course AERO 112 taught by Professor Widnall during the Fall '09 term at MIT.
- Fall '09