Simple Nature - Light and Matter

The airbag increases ∆tso as to reduce F = ∆p/∆t.less. Dividing by this much smaller ∆t gives a much larger force,perhaps thousands of newtons (with a negative sign).This is also the principle of airbags in cars. The time requiredfor the airbag to decelerate your head is fairly long: the time ittakes your face to travel 20 or 30 cm. Without an airbag, your facewould have hit the dashboard, and the time interval would havebeen the much shorter time taken by your skull to move a couple ofcentimeters while your face compressed. Note that either way, thesame amount of momentum is transferred: the entire momentum ofyour head.Force is defined as a derivative, and the derivative of a sum isthe sum of the derivatives. Therefore force is additive: when morethan one force acts on an object, you add the forces to find outwhat happens. An important special case is that forces can cancel.Consider your body sitting in a chair as you read this book. Letthe positive x axis be upward. The chair’s upward force on youis represented with a positive number, which cancels out with theearth’s downward gravitational force, which is negative. The totalrate of momentum transfer into your body is zero, and your bodydoesn’t change its momentum.Finding momentum from force example 21⊲ An object of mass m starts at rest at t = t o . A force varying asF = bt −2 , where b is a constant, begins acting on it. Find thegreatest speed it will ever have.⊲F = dpdtdp = F dt∫p = F dt + p o= − b t + p o ,where p o is a constant of integration. The given initial conditionis that p = 0 at t = t o , so we find that p o = b/t o . The negativeterm gets closer to zero with increasing time, so the maximummomentum is achieved by letting t approach infinity. That is, theobject will never stop speeding up, but it will also never surpass acertain speed. In the limit t → ∞, we identify p o as the momentumthat the object will approach asymptotically. The maximumvelocity is v = p o /m = b/mt o .Discussion QuestionA Many collisions, like the collision of a bat with a baseball, appear tobe instantaneous. Most people also would not imagine the bat and ball asbending or being compressed during the collision. Consider the followingpossibilities:146 Chapter 3 Conservation of Momentum