Principle of Impulse and Momentum

The impulse £ of a force F over a time interval At = tf — ti is the integral of the force F over the time interval:

The entity £ is a vector and has the units of N-s. If the force does not change direction during the time period At, the magnitude of the impulse is equal to the area under the ||F||-time curve (Fig. 7.1).

Figure 7.1. The plot of a magnitude of impulsive force F against time. The magnitude of the impulse £ generated by F is equal to the area under the force-time curve.

The impulse acting on an object is directly related to the change in linear momentum. The linear momentum L of an object at time t was previously defined as

in which m is the mass of the object and vc is the velocity of its center of mass at time t (Fig. 7.2). According to Newton's second law, the equation of motion of the center of mass of an object is d L/dt = 2F

where 2F denotes the resulting force acting on the object. Integrating the equation of motion for the center of mass between time t = ti and t = tf, we obtain the following relationship:

in which mvcf and mvci denote, respectively, the linear momentum of the body immediately after and before the time interval At = tf - ti, and is the linear impulse imparted to the system by external forces during that

Figure 7.2. The effect of impulse on the linear momentum of an object. The vectorial sum of the impulse and the linear momentum before the impulse must be equal to the linear momentum after the impulse.

"Tmv

period of time. According to this equation, the change in the linear momentum of a body during the time interval At is equal to the impulse acting on the body during the same time interval (Fig. 7.2). The linear momentum of the body in the direction normal to the impulse remains unaffected.

In some situations an external force acting on a body is large compared to other forces exerted on the body but the time interval during which the force acts is small. A force that becomes very large during a very small time interval is called an impulsive force. When an impulsive force acts on a body, there may be an appreciable alteration in velocity during the period of application of the force. If some of the external forces acting on the object during a time interval (tf - ti) are impulsive, we may neglect entirely the effect of all other external forces on the motion of the object in the same time interval. Although the velocities may be altered as a result of impulse, the change in the spatial position of an object is negligible.

Example 7.1. Use of Seat Belts in a Car Crash. A car traveling at a speed of 120 km/h hits a thick concrete wall. Because of the deformation of the front of the car during collision, it takes 0.5 s for the car to come to a complete stop. Determine the average impact force on a front-seat passenger who is (i) buckled and (ii) not buckled. Experments with dummies indicate that if a passenger were not to wear a seat belt, he or she would hit the windshield and that collision would take place in 1 ms. In this particular case, we assume the weight of the front-seat passenger to be 60 kg.

Solution: The case is schematically shown in Fig. 7.3. Because the impulsive force from the collision is much greater than other forces acting on the front passenger (the weight of the passenger and the contact forces between the seat and the passenger), impulse f in both cases is equal to

£ = 60 kg [0 - (120 km/h)] e1 = -2000 kg • m/s e1

ing a seat belt would decelerate as the front of the car was being deformed by the forces of impact. However, a passenger not wearing a seat belt would hit the front of the colliding car with the oncoming velocity.

in which e1 is the unit vector along the direction of motion before the crash occurs. This impulse on the passenger wearing the seatbelt occurs in 0.5 s and therefore the average crash force acting on the passenger wearing the seat belt is equal to

Fav = -2,000 (kg • m/s)/(0.5 s)] ei = -4,000 ei (N)

Let us now consider the average acceleration of the passenger wearing the seat belt during the crash. The only significant horizontal force acting on the passenger is the force of impulse by the seat belt. This force divided by the mass of the passenger must be equal to its acceleration (deceleration in this case) and thus:

This acceleration is approximately sevenfold greater than the gravitational acceleration, and studies on cadavers have shown that at this level of acceleration, the front-seat passenger might escape the crash without a significant head injury. Note that the longer it takes for the car to deform and absorb the shock, the better it is for the safety of the passenger. It is clear that cars loaded in the front with energy-absorbing materials such as steel will have lower values of average acceleration than a light and small car.

Next let us consider the front-seat passenger who is not wearing a seat belt. Even after the front of the car hits the wall and begins decelerating, because the passenger is not tied to the car, he or she will still be moving forward at 120 km/h. The passenger soon will hit the front windshield with this velocity, and the duration of the crash is 1 ms in this case. Therefore, the average crash force and the average acceleration during the crash become

This value is about 3,400 times the gravitational acceleration, and surveys of car crashes indicate significant injury to the head at acceleration values above 200 g. It would have been a miracle if the passenger had survived.

When two objects collide, the impulse forces they exert on each other are equal in magnitude but opposite in direction. Writing Eqn. 7.3 for objects A and B and then summing them, one can show that mA vAf + mB vBf = mA vA/ + mB vB,- (7.4)

in which mA and mB are the masses of objects A and B, and vA and vB denote their velocity of center of mass (Fig. 7.4). The subscripts i and f refer to times ti and tf, respectively. According to this equation, the sum of linear momentum of two objects during impact must be conserved. Impulsive force acts as an internal force for a system of colliding objects.

mB v B

mA v fA

mA v fA

m v;

Figure 7.4. The conservation of linear momentum during collision of two objects A and B. The symbols mA and mB denote, respectively, the masses of objects A and B. The vector v denotes the velocity of the center of mass of an object. The subscripts i and f refer to times immediately before and after the collision, respectively. The vector n is the unit vector that is normal to the tangent plane T at the point of contact. Note that the sense of direction of n is chosen arbitrarily.

Let us illustrate this equation with a simple example from classical mechanics. Suppose that a rigid body is dropped from a height h upon a mass-spring system (Fig. 7.5). Almost at the instant of contact the mass attached to the spring acquires momentum, the falling body loses momentum, and the two bodies begin to move as one. The conservation of momentum requires that mA vo = (mA + mB) vf

 mA >vo