Chapter 6 Momentum Can we solve conveniently all classica mechanical problems with Newtons three laws? No, the problems such as collisions
Chapter 6 Momentum Can we solve conveniently all classical mechanical problems with Newton’s three laws? No, the problems such as collisions
Non-touched collisions NICMOS· Infrared This busy image was recorded at cern(欧洲 Four galaxies colliding 粒子物理研究所),in taken by using Hubble Geneva Switzerland Space telescope
This busy image was recorded at CERN(欧洲 粒子物理研究所), in Geneva, Switzerland Four galaxies colliding taken by using Hubble Space Telescope. Non-touched collisions
6-1 How to analyze a collision? In a collision two objects exert forces on each other for an identifiable(可确认的) time interval so we can separate the motion into three parts Before, during, and after the collision During the collision, the objects exert forces on each other, these forces are equal in magnitude and opposite in direction
6-1 How to analyze a collision? In a collision, two objects exert forces on each other for an identifiable (可确认的) time interval, so we can separate the motion into three parts. Before, during, and after the collision. During the collision, the objects exert forces on each other, these forces are equal in magnitude and opposite in direction
Characteristics of a collision 1) We usually can assume that these forces are much larger than any forces n the environment. The forces vary with exerted on the two objects by other body time in a complex way 2)The time interval during the collision is quite short compared with the time during which we are watching These forces are called impulsive forces(冲力)
1) We usually can assume that these forces are much larger than any forces exerted on the two objects by other bodys in the environment. The forces vary with time in a complex way. 2) The time interval during the collision is quite short compared with the time during which we are watching. These forces are called “impulsive forces (冲力)”. Characteristics of a collision
6-2 Linear momentum o analyze collisions we define a new dynamic variable the linear momentum as P (6-1) The direction of p is the same as the direction of v The momentum p (like the velocity) depends on the reference frame of the observer and we must always specify this frame
6-2 Linear Momentum To analyze collisions, we define a new dynamic variable, the “linear momentum” as: (6-1) The direction of is the same as the direction of . The momentum (like the velocity) depends on the reference frame of the observer, and we must always specify this frame. P m v P v P
Can p be related to F? dv dmy dp ∑ F=ma=m ∑F dP (6-2) dt Any conditions for existence of above Eq. The equivalence of∑h=mand∑F= depends on the mass being a constan
F ma d P F dt The equivalence of and depends on the mass being a constant. Any conditions for existence of above Eq.? Can P be related to ? dv dmv dP F ma m dt dt dt dP F dt (6-2) F
6-3 Impulse(冲量) and momentum(动量 Fig 6-6 In this section we consider the relationship between the force that acts on a F(t) body during a collision and the change in the momentum of that body. IF Fig 6-6 shows how the magnitude of the force 0 might change with time during a co∥sio冂
6-3 Impulse(冲量) and Momentum(动量) In this section, we consider the relationship between the force that acts on a body during a collision and the change in the momentum of that body. Fig 6-6 shows how the magnitude of the force might change with time during a collision. t F F(t) Fig 6-6 Fav i t f 0 t
From Eq 6-2), we can write the change in momentum as dp=>Fdt entire collision, we integrate over the time of e To find the total change in momentum during the collision, starting at time t, (the momentum is Pi and ending at time t (the momentum isPr): 「dF=∫∑h(63)
From Eq(6-2), we can write the change in momentum as To find the total change in momentum during the entire collision, we integrate over the time of collision, starting at time (the momentum is )and ending at time (the momentum is ): (6-3) dP Fdt i t f t f f i i P t t P d P Fdt P f Pi
The left side of eq 6-3)is the change in momentum AP=Pf-Pi The right side defines a new quantity called the impulse. For any arbitrary force p, the impulse I is defined as J=「Fdt(6-4) a impulse has the same units and dimensions as momentum From Eq(6-4) and(6-3),we obtain the impulse-momentum theorem △P=Pr-P (6-5)
The left side of Eq(6-3) is the change in momentum, The right side defines a new quantity called the impulse. For any arbitrary force , the impulse is defined as (6-4) A impulse has the same units and dimensions as momentum. From Eq(6-4) and (6-3), we obtain the “ ” : (6-5) P P f Pi f i t t J F d t F J J P Pf Pi
Notes 1. Eq 6-5)is just as general as Newtons second law 2. Average impulsive force F J=Fa△t=Pr-P f 3. The external force may be negligible, compared to the impulsive force
Notes: 1. Eq(6-5) is just as general as Newton’s second law 2. Average impulsive force J Favt P f Pi F av 3. The external force may be negligible, compared to the impulsive force
