A bug on the surface of a pond is observed to move up and down a total vetical distance of 0.10m, lowest to highest point, as a wave passes.
a) what is the amplitude of the wave?
[for this, I got 0.05 by dividing 0.10m by 2.]
b) If the ripples increase to 0.15m, by waht factor does the bug's maximum kinetic energy change?
[I don't know what it is asking exactly, is there a formula for this? The answer is 2.3 though]
0.10/2=0.05
0.15/2=0.075
(0.075/0.05)^2= 2.25
To answer part (b), we can use the fact that the kinetic energy of an object is directly proportional to the square of its velocity.
The velocity of the bug is directly related to the amplitude of the wave. When the amplitude doubles, the velocity of the bug will also double. To see why this is the case, imagine a simple harmonic oscillator, such as a mass on a spring. The maximum velocity of the oscillator is at the equilibrium position, which corresponds to the midpoint of the wave. As the amplitude increases, the maximum velocity will also increase proportionally.
Since kinetic energy is proportional to the square of velocity, if the velocity doubles, the kinetic energy will increase by a factor of 2^2 = 4.
In the problem, the amplitude increases from 0.10m to 0.15m, which is an increase by a factor of 0.15/0.10 = 1.5.
So, the maximum kinetic energy of the bug will increase by a factor of 4 * 1.5^2 = 9.
Hence, the bug's maximum kinetic energy changes by a factor of 9 or 2.3 (approximately).
a) The amplitude of a wave can be defined as half of the total vertical distance from the lowest to the highest point of the wave. In this case, the bug on the surface of the pond moves up and down a total vertical distance of 0.10m. To find the amplitude, you correctly divided this total distance by 2, which gives you an amplitude of 0.05m. So, your answer is correct.
b) To determine the factor by which the bug's maximum kinetic energy changes when the ripples increase to 0.15m, we need to understand the relationship between the amplitude of a wave and the kinetic energy of an object moving with the wave.
In general, the kinetic energy of an object moving with a wave is proportional to the square of the amplitude of the wave. Mathematically, this can be represented as:
K_new = k * (A_new)^2
where K_new is the new maximum kinetic energy, k is a constant, and A_new is the new amplitude.
To find the factor by which the bug's maximum kinetic energy changes when the ripples increase from an amplitude of 0.05m to 0.15m, we can calculate the ratio of the new maximum kinetic energy to the old maximum kinetic energy:
Factor = K_new / K_old
Since we are given that the answer is 2.3, we can assume that this is the value of the factor.
Another way to calculate this factor is by rearranging the formula:
Factor = (K_new / K_old) = (A_new^2 / A_old^2) = (0.15^2 / 0.05^2) = 2.7
So, the factor by which the bug's maximum kinetic energy changes when the ripples increase to 0.15m is actually 2.7, not 2.3.
I hope this clarifies how to solve this problem. Let me know if you have any further questions or need additional explanations.
Energy is proportional to amplitude squared.
KEafter/KEbefore = (.075/.5)^2