If the displacement of a moving point at any time is givenby an equation of the form `y (t) = a cos omega t + b sin omega t`, shown that the motion is simple harmonic . If `a = 3 m, b = 4m and omega = 2`: determine the period , amplitude, maximum velocity and maximum acceleration.

Explanation

To solve the problem step by step, we will first show that the motion described by the equation \( y(t) = a \cos(\omega t) + b \sin(\omega t) \) is simple harmonic motion (SHM). Then, we will calculate the period, amplitude, maximum velocity, and maximum acceleration given \( a = 3 \, \text{m} \), \( b = 4 \, \text{m} \), and \( \omega = 2 \).

### Step 1: Proving the Motion is Simple Harmonic
The general form of SHM can be expressed as:
\[ y(t) = A \sin(\omega t + \phi) \]
where \( A \) is the amplitude and \( \phi \) is the phase constant.

We can rewrite the given equation \( y(t) = a \cos(\omega t) + b \sin(\omega t) \) in the form of SHM. To do this, we can find the amplitude \( A \) and phase angle \( \phi \) using the following relationships:
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