The potential energy for a mss m = 1.0 kg moving in one dimension is given by U(x) = (2.5J) sin πx.
The mass starts at x = 0 with an initial velocity v = 0.71m/s. (The plus sign means the motion is
in the positive x-direction.) Describe the subsequent motion of the mass. Suppose the initial velocity
were 3.0 m/s. What would the subsequent motion look like?
The Attempt at a Solution
Looking at the initial conditions when x=0, U(x)=0.
Considering the initial velocity as 0.71m/s, substituting this and the mass into Ek=1/2mv2 we obtain
We know that sin maximizes at 1/2 pi, thus, at x=0.5 our U(x) function will be maximized.
U(0.5)=(2.5J)sin π(0.5) = 2.5 J
Since Ek=0.25205 J
So, Ek<U(0.5) and we conclude that the object will not reach the top of the peak defined by our Potential energy function.
Moving ahead early (I'll explain why in a moment)
for part b, we see that Ek=1/2(1kg)(3m/s)2
so Ek=4.5 J and therefore now Ek>U(0.5) so it will pass over the crest.
My question for this problem is as follows:
In part a), will the object fall into negative values of x? and therefore also negative potential energy?
In part b) will the object continue over the crest and again, proceed into negative potential energy?
I recall my professor explaining that potential energy is set by the person, so negative values are able to be obtained. In this situation, I see part a oscillating back and forth, while in part b it continues on with its motion indefinitely.
What is the best way to "Describe the subsequent motion". This seems like a loose statement with a fair amount of points attached to it vulnerable to misinterpretation on my part.
Thank you for any insight to this problem.