Physics
Momentum Change and Impulse
Momentum change refers to the alteration in an object's momentum, which is the product of its mass and velocity. Impulse, on the other hand, is the force applied to an object over a period of time, resulting in a change in its momentum. These concepts are important in understanding how forces affect the motion of objects.
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9 Key excerpts on "Momentum Change and Impulse"
eBook - PDF- Michael Tammaro(Author)
- 2019(Publication Date)
- Wiley(Publisher)
The linear momentum p of an object is equal to the product of its mass m and its velocity v: = m p v (7.1.2) The SI unit of momentum is / kg m s ⋅ . Like impulse, linear momentum is a vector quantity, and an object’s momentum has the same direction as its velocity. 7.2 The Impulse–Momentum Theorem The change in an object’s momentum Dp is equal to the product of the average net force acting on the object F avg ∑ and the time interval Δt over which that force acts: ∑ ( ) D = D t F p avg (7.2.2) Equation 7.2.2 is called the impulse–momentum theorem, which can be stated in words as follows: The impulse imparted by the average net force acting on an object is equal to the change in the object’s momentum. The impulse–momentum theorem is simply a restate- ment of Newton’s second law. 7.3 Conservation of Momentum In physics, a system is collection of interacting entities (particles, objects, etc.) that is cho- sen for analysis. If the net external force on a system of objects is zero, then the total linear momentum of that system does not change: D = P 0 or = P P i f (7.3.3) Equation 7.3.3 is the mathematical statement of conservation of linear momentum, where capital “P” is used to denote the total momentum of the system. 7.4 One-Dimensional Collisions A collision is an interaction between two or more objects that results in a change in the momentum of the objects (but not necessarily the total momentum). Collisions can be classified as either elastic or inelastic. In physics, an elastic collision is one in which the total kinetic energy of the system is the same before and after the collision. Ine- lastic collisions, on the other hand, are those in which the kinetic energy of the system changes. In a completely inelastic collision, the objects do not “rebound” from one another—they stick together and essentially become a single physical object. If the 202 | Chapter 7 colliding objects constitute an isolated system, then linear momentum is conserved.
eBook - PDF- Richard C. Hill, Kirstie Plantenberg(Authors)
- 2013(Publication Date)
- SDC Publications(Publisher)
76 CHAPTER 9 ACTIVITIES ............................................................................................... 77 Conceptual Dynamics Kinetics: Chapter 9 – Particle Impulse & Momentum 9 - 2 CHAPTER SUMMARY In this chapter, we study the last of the three approaches we will employ for analyzing the kinetics of particles: the impulse-momentum method. Previously, we studied Newtonian mechanics and the work-energy method. The impulse-momentum method involves examining the cumulative effect of forces and moments over time and how they affect a body's motion. There are two applications of this methodology, linear impulse- momentum and angular impulse-momentum. The linear impulse-momentum approach relates forces, time, linear velocities, and masses, while the angular impulse-momentum approach relates moments, time, angular velocities, and mass moments. Often, more than one of the three kinetic analysis methods: Newtonian mechanics, work-energy, and impulse-momentum can be used to analyze a specific situation. However, selecting the "best" approach can minimize the amount of calculation. It can also be the case that the required information is more readily available (i.e. can be measured or estimated more easily) for a specific analysis method. Conceptual Dynamics Kinetics: Chapter 9 – Particle Impulse & Momentum 9 - 3 9.1) LINEAR MOMENTUM We have all used the term "momentum" in our everyday lives. You are working on a task and you don't want to stop because you will lose momentum. Or, a sports team is on a winning streak so they have momentum and they are going to be hard to stop. These two examples, in a sense, capture the meaning of the precise mechanics/physics definition of momentum. These examples give you a sense that you and the team have some sort of forward motion that will carry you through your next obstacle. The more momentum you have, the more difficult it will be for you to stop. Momentum, as defined in mechanics, depends on mass and velocity.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
Definition of Impulse The impulse J B of a force is the product of the average force F B and the time interval Dt during which the force acts: J B 5 F B D t (7.1) Impulse is a vector quantity and has the same direction as the average force. SI Unit of Impulse: newton ? second (N ? s) When a ball is hit, it responds to the value of the impulse. A large impulse produces a large response; that is, the ball departs from the bat with a large velocity. However, we know from experience that the more massive the ball, the less velocity it has after leaving the bat. Both mass and velocity play a role in how an object responds to a given impulse, and the effect of each of them is included in the concept of linear momentum, which is defined as follows: Definition of Linear Momentum The linear momentum p B of an object is the product of the object’s mass m and velocity v B : p B 5 mv B (7.2) Linear momentum is a vector quantity that points in the same direction as the velocity. SI Unit of Linear Momentum: kilogram ? meter/second (kg ? m/s) Newton’s second law of motion can now be used to reveal a relationship between impulse and momentum. Figure 7.3 shows a ball with an initial velocity v 0 B approaching a bat, being struck by the bat, and then departing with a final velocity v f B . When the velocity of an object changes from v 0 B to v f B during a time interval Dt, the average acceleration a B is given by Equation 2.4 as a B 5 v B f 2 v B 0 D t According to Newton’s second law, SF B 5 m a B , the average acceleration is produced by the net average force SF B . Here SF B represents the vector sum of all the average forces that act on the object. Thus, SF B 5 m a v B f 2 v B 0 D t b 5 mv B f 2 mv B 0 D t (7.3) In this result, the numerator on the far right is the final momentum minus the initial mo- mentum, which is the change in momentum.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
Definition of Impulse The impulse J B of a force is the product of the average force F B and the time interval Dt during which the force acts: J B 5 F B D t (7.1) Impulse is a vector quantity and has the same direction as the average force. SI Unit of Impulse: newton ? second (N ? s) When a ball is hit, it responds to the value of the impulse. A large impulse produces a large response; that is, the ball departs from the bat with a large velocity. However, we know from experience that the more massive the ball, the less velocity it has after leaving the bat. Both mass and velocity play a role in how an object responds to a given impulse, and the effect of each of them is included in the concept of linear momentum, which is defined as follows: Definition of Linear Momentum The linear momentum p B of an object is the product of the object’s mass m and velocity v B : p B 5 mv B (7.2) Linear momentum is a vector quantity that points in the same direction as the velocity. SI Unit of Linear Momentum: kilogram ? meter/second (kg ? m/s) Newton’s second law of motion can now be used to reveal a relationship between impulse and momentum. Figure 7.3 shows a ball with an initial velocity v 0 B approaching a bat, being struck by the bat, and then departing with a final velocity v f B . When the velocity of an object changes from v 0 B to v f B during a time interval Dt, the average acceleration a B is given by Equation 2.4 as a B 5 v B f 2 v B 0 D t According to Newton’s second law, SF B = m a B , the average acceleration is produced by the net average force SF B . Here SF B represents the vector sum of all the average forces that act on the object. Thus, SF B 5 m a v B f 2 v B 0 D t b 5 mv B f 2 mv B 0 D t (7.3) In this result, the numerator on the far right is the final momentum minus the initial mo- mentum, which is the change in momentum.
- David Halliday, Jearl Walker, Patrick Keleher, Paul Lasky, John Long, Judith Dawes, Julius Orwa, Ajay Mahato, Peter Huf, Warren Stannard, Amanda Edgar, Liam Lyons, Dipesh Bhattarai(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
That change is related to the force by Newton’s second law written in the form F = d p∕dt. By rearranging this second‐law expression, we see that, in time interval dt, the change in the ball’s momentum is d p = F (t) dt. (9.20) We can find the net change in the ball’s momentum due to the collision if we integrate both sides of this equation: ∫ p 2 p 1 d p = ∫ t 2 t 1 F (t) dt. (9.21) The left side of this equation gives us the change in the momentum: p f − p i = Δ p. The right side, which is a measure of both the magnitude and the duration of the collision force, is called the impulse J of the collision: J = ∫ t 2 t 1 F (t) dt. (9.22) Pdf_Folio:150 150 Fundamentals of physics Thus, the change in an object’s momentum is equal to the impulse on the object: Δ p = J. (9.23) This expression, which is known as the impulse–linear momentum theorem, can also be written in the vector form p f − p i = J (9.24) and in component forms such as Δp x = J x (9.25) and p fx − p ix = ∫ t f t i F x dt. (9.26) Integrating the force If we have a function for F (t), we can evaluate J (and thus the change in momentum) by integrating the function. If we have a plot of F versus time t, we can evaluate J by finding the area between the curve and the t axis, such as in figure 9.9. In many situations we do not know how the force varies with time but we do know the average magnitude F avg of the force and the duration Δt (= t f − t i ) of the collision. Then we can write the magnitude of the impulse as J = F avg Δt. (9.27) The average force is plotted versus time as shown in figure 9.10. FIGURE 9.9 The curve shows the magnitude of the time-varying force F(t) that acts on the ball in the collision of fgure 9.8. The area under the curve is equal to the magnitude of the impulse J on the ball in the collision. t i F J F(t) t f Δ t t The impulse in the collision is equal to the area under the curve.
eBook - PDF- Robert Resnick, David Halliday, Kenneth S. Krane(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
We could, for example, use Eq. 6-5 to find the momentum acquired by a body falling in the Earth’s gravity. We defined the impulse in terms of a single force, but the impulse-momentum theorem deals with the change in momentum due to the impulse of the net force — that is, the combined effect of all the forces that act on the particle. In the case of a collision involving two particles, there is often no distinction because each particle is acted upon by only one force, which is due to the other particle. In this case, the change in momentum of one particle is equal to the im- pulse of the force exerted by the other particle. The impulsive force whose magnitude is represented in Fig. 6-6 is assumed to have a constant direction. The mag- J net,z p z p fz p iz . J net,y p y p fy p iy , J net, x p x p f x p i x , J B net p B p B f p B i . J B net F B dt. J B t f t i F B dt. J B F B , p B p B f p B i . p B f p B i d p B t f t i F B dt. p B f ): p B i ) d p B F B dt. F B nitude of the impulse of this force is represented by the area under the F(t) curve. We can represent that same area by the rectangle in Fig. 6-6 of width t and height F av , where F av is the magnitude of the average force that acts during the interval t. Thus (6-7) In a collision such as that of the ball and bat of Fig. 6-1, it is difficult to measure F(t) directly, but we can estimate t (perhaps a few milliseconds) and obtain a reasonable value for F av based on the impulse computed according to Eq. 6-6 from the change in momentum of the ball (see Sample Problem 6-1). We have defined a collision as an interaction that occurs in a time t that is negligible compared to the time during which we are observing the system. We can also character- ize a collision as an event in which the external forces that may act on the system during the time of the collision are negligible compared to the impulsive collision forces.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler, Heath Jones, Matthew Collins, John Daicopoulos, Boris Blankleider(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
As a result, the ball’s velocity changes from an initial value of v 0 (top drawing) to a final value of v f (bottom drawing). v f v 0 F Newton’s second law of motion can now be used to reveal a relationship between impulse and momen- tum. Figure 7.3 shows a ball with an initial velocity v 0 approaching a bat, being struck by the bat, and then departing with a final velocity v f . When the velocity of an object changes from v 0 to v f during a time interval Δt, the average acceleration a is given by equation 2.4 as a = v f - v 0 Δt According to Newton’s second law, Σ F = m a the average acceleration is produced by the net average force Σ F. Here Σ F represents the vector sum of all the average forces that act on the object. Thus, Σ F = m ( v f - v 0 Δt ) = m v f - m v 0 Δt (7.3) In this result, the numerator on the far right is the final momentum minus the initial momentum, which is the change in momentum. Thus, the net average force is given by the change in momentum per unit of time. * Multiplying both sides of equation 7.3 by Δt yields equation 7.4, which is known as the impulse– momentum theorem. *The equality between the net force and the change in momentum per unit time is the version of the second law of motion presented originally by Newton. CHAPTER 7 Impulse and momentum 173 Impulse–momentum theorem When a net average force Σ F acts on an object during a time interval Δt the impulse of this force is equal to the change in momentum of the object: (Σ F)Δt ⏟⏞⏞ ⏟⏞⏞ ⏟ Impulse = m v f ⏟ ⏟ ⏟ Final momentum - m v 0 ⏟ ⏟ ⏟ Initial momentum (7.4) Impulse = Change in momentum During a collision, it is often difficult to measure the net average force Σ F, so it is not easy to determine the impulse (Σ F)Δt directly. On the other hand, it is usually straightforward to measure the mass and velocity of an object, so that its momentum m v f just after the collision and m v 0 just before the collision can be found.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
LEARNING OBJECTIVES After reading this module, you should be able to... 7.1 Apply the impulse–momentum theorem. 7.2 Apply the law of conservation of linear momentum. 7.3 Analyze one-dimensional collisions. 7.4 Analyze two-dimensional collisions. 7.5 Determine the location and the velocity of the center of mass. Stephan Goerlich/Age Fotostock America CHAPTER 7 Impulse and Momentum In the sport of jousting in the Middle Ages, two knights in armor rode their horses toward each other and, using their lances, attempted to knock each other to the ground. Sometimes, however, the collision between a lance and an opponent’s shield caused the lance to shatter and no one was unseated. In physics such a collision is classified as being inelastic. Inelastic collisions are one of the two basic types that this chapter introduces. 7.1 The Impulse–Momentum Theorem There are many situations in which the force acting on an object is not constant, but varies with time. For instance, Figure 7.1a shows a baseball being hit, and part b of the figure illustrates approximately how the force applied to the ball by the bat changes during the time of contact. The magnitude of the force is zero at the instant t 0 just before the bat touches the ball. During contact, the force rises to a maximum and then returns to zero at the time t f when the ball leaves the bat. The time interval ∆t = t f ‒ t 0 during which the bat and ball are in contact is quite short, being only a few thousandths of a second, although the maximum force can be very large, often exceeding thousands of newtons. For comparison, the graph also shows the magnitude F of the average force exerted on the ball during the time of contact. Figure 7.2 depicts other situations in which a time-varying force is applied to a ball. To describe how a time-varying force affects the motion of an object, we will introduce two new ideas: the impulse of a force and the linear momentum of an object.
eBook - PDFA Course of Mathematics for Engineers and Scientists
Volume 3: Theoretical Mechanics
- Brian H. Chirgwin, Charles Plumpton(Authors)
- 2014(Publication Date)
- Pergamon(Publisher)
At the beginning of this interval all these different parts of the system have their separate states of motion; at the end of the interval each part has, usually, a different state of motion. Equation (10.8) states that the change of linear momentum of the whole system in the interval ôt is given by the impulse of the external forces acting on the whole system. Having written down the equation express-ing this momentum balance for an interval of time ôt, we obtain a differential equation by dividing by ôt and then taking the limit as ôt -+ 0. In Chap. VII p. 238 we pointed out the effect 'an intelligent being' could have on a system by the use of its muscles. The process con-sidered here is a generalisation of this idea. 1. A body falls under constant gravity picking up matter from rest as it falls, so that at time t its mass is m and speed v. Consider the system which consists of the body at a given instant and the material, of mass ôm, which it picks up in the subsequent interval ôt; this material is initially at rest. The initial momentum of this system is p = mv + ôm · 0. Finally the body and the additional material are moving with velocity v + ôv, and the final momentum is p + ôp — (m + ôm) (v + ôv). The external force acting on the whole system throughout this interval is the total weight (m + ôm)g. .'. (m + ôm)g ôt = δρ = (m + ôm) (v + ôv) — mv. (10.12) Correct to the first order this gives mg ôt = v ôm + m ôv. This leads to the differential equation, in the limit as ôt -> 0, àm άν d mg = v ^r + m ^j = i ü <»«>· ( 1 0 · 1 3 ) § 10:5 I M P U L S I V E MOTION AND VARIABLE MASS 355 If the matter picked up had been moving downwards with speed u, then in place of eqn. (10.12), the momentum balance would have been (m + ôm)g ôt = ôp = (m + ôm) (v + ôv) — (mv + ôm · w). dv , dm dm d dm ,„ Λ , .. ·'· w ^ = m -cû + î; -dr-M ^r = w(^)-M ^-· (1014) 2. A body moves vertically upwards under gravity so that at time t its mass is m and its speed is v.
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