i/Two balls start from rest,
and roll from A to B by different
paths.
field. If the plane can start from 10 km up, what is the maximum
amount of time for which the dive can last?
.Based on data about acceleration and distance, we want to
find time. Acceleration is the second derivative of distance, so if
we integrate the acceleration twice with respect to time, we can
find how position relates to time. For convenience, let’s pick a
coordinate system in which the positiveyaxis is down, soa=g
instead of−g.a=g
v=gt+ constant (integrating)
=gt (starts from rest)y=1
2
gt^2 + constant (integrating again)Choosing our coordinate system to havey = 0 att= 0, we can
make the second constant of integration equal zero as well, sot=√
2 y
g=
√
2 ·10000 m
10 m/s^2
=√
2000 s^2
= 40 s (to one sig. fig.)Note that if we hadn’t converted the altitude to units of meters,
we would have gotten the wrong answer, but we would have been
alerted to the problem because the units inside the square root
wouldn’t have come out to be s^2. In general, it’s a good idea to
convert all your data into SI (meter-kilogram-second) units before
you do anything with them.High road, low road example 9
.In figure i, what can you say based on conservation of energy
about the speeds of the balls when the reach point B? What does
conservation of energy tell you about which ball will get there
first? Assume friction doesn’t convert any mechanical energy to
heat or sound energy.
.Since friction is assumed to be negligible, there are only two
forms of energy involved: kinetic and gravitational. Since both
balls start from rest, and both lose the same amount of gravi-
tational energy, they must have the same kinetic energy at the
end, and therefore they’re rolling at the same speed when they
reach B. (A subtle point is that the balls have kinetic energy both
because they’re moving through space and because they’re spin-
ning as they roll. These two types of energy must be in fixed84 Chapter 2 Conservation of Energy