Question

Equations y = 2a(cos wt)^2 and y= a(sin wt + √3 cos wt ) represent the motion of two particle then find the ratio of their max speed

Answer 1

The ratio of the max velocity of the two wave functions = 4:1

Maximum velocity (V max) for a wave function is equal to product of its amplitude (a) and frequency (w).

V max = amplitude × frequency = aw

Simplifying , y = 2a(cos wt)² to general form to get the value of amplitude and frequency

=> y = 2a((cos 2w + 1)/2)

=> y = a(1+cos 2w)

(V max)1 = a × 2w

= 2 aw

Simplifying , y= a(sin wt+√3 coswt) to general form to get the value of amplitude and frequency

=> y = a/2(1/2 sin wt + √3/2 cos wt)

=> y = a/2 (cos π/3 sin wt + sin π/3 cos wt)

=> y = a/2 sin(wt + π/3)

(V max)2 = a/2 × w

= aw/2

Ratio the max speed =

(V max)1/(V max)2

= 2aw/(aw/2)

= 4/1

Ratio = 4:1

Answer 2

Answer:

Step-by-step explanation:

y = 2A cos^2wt

y= A( cos2wt + 1)

y - A = Acos2wt

So here amplitude is A ,

and w' = 2w so V max = 2Aw

Now

y= A( sinwt + √3coswt)

Y = 2Asin( wt+ π /3 )

So , here amplitude =2A

And w' = w

So ( Vmax)2 = 2Aw

So ratio is 2/2 =1 :1

So,

The ans is 1:1