A particle executes simple harmonic motion about `x=0` along x-axis. The position of the particle at an instant is given by `x= ( 5 cm) sin pi t`. Find the average velocity and average acceleration for a time interval `0-0.5 s`.
Hint `:` Average velocity `=( x_(2)- x_(1))/(t_(2)-t_(1))`
`x_(1) = 5 sin 0`
`x_(2) = 5 sin ( pi xx 0.5)`
`t_(2) - t_(1) = 0.5`
Average acceleration `= ( v_(2) - v_(1))/( t_(2) - t_(1))`
`v=(dx)/(dt) = 5 pi cos pit`
`implies a _(av) = ( 5 pi cos ( pi xx 0.5) -5 pi cos theta)/(0.5)`

Explanation

To solve the problem, we will calculate the average velocity and average acceleration of a particle executing simple harmonic motion described by the equation \( x = 5 \sin(\pi t) \) over the time interval \( t = 0 \) to \( t = 0.5 \) seconds.

### Step 1: Calculate the position at \( t_1 = 0 \) seconds
Using the position function:
\[ x_1 = 5 \sin(\pi \cdot 0) = 5 \sin(0) = 0 \, \text{cm} \]

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