Simple Nature - Light and Matter

elated to gravitational mass, but since these two masses are equal,we were able to use a single symbol, m, for them, and cancel themout.We can see from the equation v = √ −2g∆y that a falling object’svelocity isn’t constant. It increases as the object drops farther andfarther. What about its acceleration? If we assume that air frictionis negligible, the arguments in section 1.2 show that the accelerationcan’t depend on the object’s mass, so there isn’t much else theacceleration can depend on besides g. In fact, the acceleration of afalling object equals −g (in a coordinate system where the positivey axis points up), as we can easily show using the chain rule:( dvdt) ( ) ( dv dK=dK dU( 1=mv= −g ,) ( dUdy)(−1)(mg)(v)) ( dydtwhere I’ve calculated dv/ dK as 1/(dK/ dv), and dK/ dU = −1 canbe found by differentiating K+U = (constant) to give dK+dU = 0. 7We can also check that the units of g, J/kg·m, are equivalent tothe units of acceleration,Jkg·m = kg·m2 /s 2kg·m = m s 2 ,and therefore the strength of the gravitational field near the earth’ssurface can just as well be stated as 10 m/s 2 .Speed after a given time example 7⊲ An object falls from rest. How fast is it moving after two seconds?Assume that the amount of energy converted to heat byair friction is negligible.⊲ Under the stated assumption, we have a = −g, which can beintegrated to give v = −gt + constant. If we let t = 0 be thebeginning of the fall, then the constant of integration is zero, so att = 2 s we have v = −gt = −(10 m/s 2 ) × (2 s) = 20 m/s.The Vomit Comet example 8⊲ The U.S. Air Force has an airplane, affectionately known as theVomit Comet, in which astronaut trainees can experience simulatedweightlessness. Oversimplifying a little, imagine that theplane climbs up high, and then drops straight down like a rock.(It actually flies along a parabola.) Since the people are fallingwith the same acceleration as the plane, the sensation is just likewhat you’d experience if you went out of the earth’s gravitational7 There is a mathematical loophole in this argument that would allow theobject to hover for a while with zero velocity and zero acceleration. This pointis discussed on page 910.)Section 2.1 Energy 83