Chapter 4 - UCSB HEP

WORK AND ENERGY
Alternatively, we can use indefinite upper limits in Eq. (4.3):
+mv2 - +mva2 = 1%: F(I) dr,
where v is the speed of the particle when it is at position z. Equation
(4.4) gives us v as a function of x. Since v = dxldt, we could
solve Eq. (4.4) for dsJdt and integrate again to find x(t). Rather
than write out the general formula, it is easier to see the method
by studying a few examples.
Example 4.1
Mass Thrown Upward in a Uniform Gravitational Field
A mass m is thrown vertically upward with initial speed vo. How high
does it rise, assuming the gravitations! force to be constant, and neglecting
air friction
Taking the z a ~is to be directed vertically upward,
Equation (4.3) gives
At the peak, vl = O and we obtain the answer
It is interesting to note that the solution makes no reference to time
at all. We could have solved the problem by applying Newton's second
law, but we would have had to eliminate f to obtain the result.
Here is an example that is not easy to solve by direct application
of Newton's second law.
Example 4.2
Solving the Equation of Simple Harmonic Motion
In Example 2.17 we discussed the equation of simple harmonic motion
and pulled the solution out of a hat without proof. Now we shall derive
the solution using Eq. (4.4).