An oceanographer measured an ocean
wave during a storm. The vertical
displacement, h, of the wave,
in metres, can be modelled by
h(t) = 0.8cost + 0.5sin2t, where t is the
time in seconds.
a) Determine the vertical displacement
of the wave when the velocity is
0.8 m/s.
b) Determine the maximum velocity of
the wave and when it fi rst occurs.
c) When does the wave fi rst change
from a “hill” to a “trough”? Explain.
h(t) = 0.8cost + 0.5sin2t
v(t) = -.8sint + .5cos(2t)(2) = -.8sint + cos(2t)
= .8
-.8sint + cos(2t) = .8
time -10
8sint - 10cos(2t) = -8
4sint - 5cos(2t) = -4 , but cos(2t) = 2sin^2 t - 1
4sint - 5(2sin^2 t - 1) + 8 = 0
4sint - 10sin^2 t + 5 + 8 = 0
10sin^2 t - 4sint - 13 = 0
sint = (4 ± √536)/20 = 1.35 or -.95759... , but sint ≥ 0
sint = -.95759..
t = 1.2785 + π = appr 4.42 seconds
plug that into h(t) to get the displacement
b) take the derivative of v(t), set it equal to zero and solve
c) your turn, use the properties of what happens at "hill" and "troughs"
Here is a picture of h(t)
http://www.wolframalpha.com/input/?i=plot+y+%3D+0.8cosx+%2B+0.5sin(2x)
a) To determine the vertical displacement of the wave when the velocity is 0.8 m/s, we need to find the time(s) at which the velocity is 0.8 m/s and substitute it into the equation h(t) = 0.8cost + 0.5sin2t.
Now, the velocity of the wave is given by the derivative of the displacement function with respect to time, which is:
v(t) = -0.8sint - sin2t
Setting v(t) = 0.8 m/s, we have:
-0.8sint - sin2t = 0.8
To solve this equation, we can rearrange it as:
-0.8sint = 0.8 + sin2t
Divide both sides by -0.8:
sint = -(0.8 + sin2t) / 0.8
Using trigonometric identity sin2t = 1 - cos^2t, we can rewrite the equation as:
sint = -(0.8 + (1 - cos^2t)) / 0.8
Simplify:
sint = -1.25 + 0.625cos^2t
To solve this equation, you can use a calculator or a computer program to find the values of t for which sint is equal to -1.25 + 0.625cos^2t. Substitute the found values of t into the equation h(t) = 0.8cost + 0.5sin2t to find the vertical displacement at those times.
b) To determine the maximum velocity of the wave and when it first occurs, we need to find the maximum value(s) of the velocity function v(t) = -0.8sint - sin2t.
Differentiating v(t) with respect to t gives us:
v'(t) = -0.8cost - 2cos2t
Setting v'(t) = 0, we have:
-0.8cost - 2cos2t = 0
To solve this equation, we can rearrange it as:
-0.8cost = 2cos2t
Divide both sides by -0.8:
cost = -(2cos2t) / 0.8
Using trigonometric identity cos2t = 2cos^2t - 1, we can rewrite the equation as:
cost = -(2(2cos^2t - 1)) / 0.8
Simplify:
cost = -5cos^2t + 2
To solve this equation, you can use a calculator or a computer program to find the values of t for which cost is equal to -5cos^2t + 2. Substitute the found values of t into the velocity function v(t) = -0.8sint - sin2t to find the maximum velocity at those times.
c) The wave changes from a "hill" to a "trough" when its displacement changes from positive to negative. In other words, when the wave goes from above the equilibrium position to below the equilibrium position. Mathematically, this happens when the displacement function h(t) crosses the x-axis.
To find when the wave first changes from a "hill" to a "trough," we need to find the roots of the displacement function h(t) = 0.8cost + 0.5sin2t. These are the values of t for which h(t) = 0.
To solve the equation 0.8cost + 0.5sin2t = 0, you can use a calculator or a computer program to find the values of t for which this equation is true. These will give you the times at which the wave changes from a "hill" to a "trough".
To determine the vertical displacement of the wave when the velocity is 0.8 m/s, we need to find the time at which the velocity is 0.8 m/s and then substitute that time into the equation for h(t).
To find the time when velocity is 0.8 m/s, we need to find the derivative of h(t), which represents the velocity of the wave. The derivative of h(t) with respect to t can be found by differentiating each term of the equation separately.
The derivative of 0.8cost is -0.8sin(t) and the derivative of 0.5sin2t is cos(2t). Adding the two derivatives together gives us the overall derivative of h(t) as -0.8sin(t) + cos(2t).
Now we can set this derivative equal to 0.8 and solve for t.
-0.8sin(t) + cos(2t) = 0.8
To solve this equation, we can use algebraic methods or a graphing calculator that can find the intersection points of two functions.
Once we find the values of t where the velocity is 0.8 m/s, we can substitute these values into the equation h(t) = 0.8cost + 0.5sin2t to find the corresponding vertical displacements.
For part (b), to determine the maximum velocity of the wave and when it first occurs, we need to find the maximum values of the derivative of h(t) and the corresponding times.
Using the derivative we found earlier (-0.8sin(t) + cos(2t)), we can find the maximum by finding the critical points where the derivative equals 0 and checking whether these points are maximum or minimum points. Similar to part (a), we can solve this equation to find the critical points.
Once we find the critical points, we can substitute these values into the derivative to determine whether they are maximum or minimum points. The maximum point will represent the maximum velocity of the wave, and the time at which it occurs can be found by substituting the critical point into the equation for h(t).
For part (c), we need to determine when the wave changes from a "hill" to a "trough." This occurs when the sign of the derivative changes from positive to negative. In other words, when the slope of the wave changes from increasing to decreasing.
To find the time when the wave changes from a "hill" to a "trough," we need to find where the derivative changes sign. This can be done by finding the critical points where the derivative equals 0 and then checking the sign of the derivative on either side of these points.
If the sign of the derivative changes from positive to negative at a critical point, that point represents the time at which the wave changes from a "hill" to a "trough." Again, we can substitute this time into the equation for h(t) to find the corresponding vertical displacement.
By following these steps, we can determine the answers to parts (a), (b), and (c) of the given question.