Answer:
a) 5 m
b) 0.667 s
c) 0.5 m
d) 0.75 m/s
e) -5 m
Explanation:
In an equation of the form
y(x, t) = Acos(kx - ωt)
A is the amplitude, ω = 2π/T where T is the period, and k = 2π/λ where λ is the wavelength. In this case, the equation os
y(x,t) = 5cos(π(4x - 3t)
y(x,t) = 5cos(4πx - 3πt)
So, A = 5, k = 4π, and ω = 3π. Then, we can find each part as follows
a) Amplitude
The amplitude is A, so it is 5 m.
b) the period
Using the equation ω = 2π/T and solving for T, we get:
[tex]T=\frac{2\pi}{\omega}=\frac{2\pi}{3\pi}=\frac{2}{3}=0.667\text{ s}[/tex]So, the period is 0.667 s
c) the wavelength.
using the equation k = 2π/λ and solving for λ, we get:
[tex]\lambda=\frac{2\pi}{k}=\frac{2\pi}{4\pi}=0.5\text{ m}[/tex]So, the wavelength is 0.5 m
d) The wave speed
The wave speed can be calculated as the division of the wavelength by the period, so
[tex]v=\frac{\lambda}{T}=\frac{0.5\text{ m}}{0.667\text{ s}}=0.75\text{ m/s}[/tex]e) The height of the wave at (2, 1)
To find the height, we need to replace (x, t) = (2, 1) on the initial equation, so
[tex]\begin{gathered} y(x,t)=5\cos(\pi(4x-3t)) \\ y(2,1)=5\cos(\pi(4\cdot2-3\cdot1)) \\ y(2,1)=5\cos(\pi(8-3)) \\ y(2,1)=5\cos(\pi(5)) \\ y(2,1)=5\cos(5\pi) \\ y(2,1)=5(-1) \\ y(2,1)=-5 \end{gathered}[/tex]Then, the height of the wave is -5 m.
Therefore, the answers are
a) 5 m
b) 0.667 s
c) 0.5 m
d) 0.75 m/s
e) -5 m
