If x denotes displacement in time t and x=a cost, then acceleration is :-

Options

A. a cos t
B. –a cos t
C. a sin t
D. – a sin t

Answer

To find the acceleration when the displacement \( x \) is given by the equation \( x = a \cos t \), we will follow these steps:

### Step 1: Differentiate the displacement to find velocity
The first step is to differentiate the displacement \( x \) with respect to time \( t \) to find the velocity \( v \).

\[ v = \frac{dx}{dt} = \frac{d}{dt}(a \cos t) \]

### Step 2: Apply the differentiation
Since \( a \) is a constant, we can take it out of the differentiation. The derivative of \( \cos t \) is \( -\sin t \).

\[ v = a \cdot (-\sin t) = -a \sin t \]

### Step 3: Differentiate the velocity to find acceleration
Next, we differentiate the velocity \( v \) with respect to time \( t \) to find the acceleration \( a \).

\[ a = \frac{dv}{dt} = \frac{d}{dt}(-a \sin t) \]

### Step 4: Apply the differentiation again
Again, since \( a \) is a constant, we can take it out. The derivative of \( -\sin t \) is \( -\cos t \).

\[ a = -a \cdot \cos t \]

### Final Result
Thus, the acceleration is given by:

\[ a = -a \cos t \]

### Conclusion
The correct expression for acceleration when \( x = a \cos t \) is \( -a \cos t \).

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