Solution:
x = 10sin 2t + 15cos 2t +100
v = dx/dy = 20cos 2t - 30sin 2t
a = dv/dt = 40sin 2t - 60cos 2t
For trigonometric function set calculator to radian:
(a) At t = 1 s x1 = 10sin 2 + 15cos 2 + 100 = 102.9
v1 = 20cos 2 - 30sin 2 = -35.6
a1 = -40sin 2 - 60cos 2 = -11.40
(b) Maximum velocity occure when a = 0.
-40sin 2t - 60cos 2t = 0
tan 2t = -60/40 = -1.5
2t = tan⁻¹ (-1.5) = -0.9828 and -0.9828 + [tex]\pi[/tex]
Reject the negative value. 2t = 2.1588
t = 1.0794[tex]s[/tex]
t = 1.0794[tex]s[/tex] for V[tex]max[/tex]
so [tex]Vmax[/tex] =20cos(2.1588) -30sin(2.1588)
= -36.056
Note that we could have also used
[tex]Vmax[/tex] = [tex]\sqrt[/tex]20² +30²
= 36.056
by combining the sine and cosine terms.
For [tex]\alpha max[/tex] we can take the derivative and set equal to zero or just combine the sine and cosine terms.
[tex]\alpha max[/tex] = [tex]\sqrt{40^{2} + 60^{2}[/tex]
= 72.1 mm/[tex]s^{2}[/tex]
