The position x of a body is defined by equation `x = Pt^(2)- Q t^(3)`.
The acceleration of the particle will be zero at time equal to

Options

A. `(P)/(Q)`
B. `(Q)/(P)`
C. `(P)/(3Q)`
D. `(3P)/(Q)`

Explanation

To find the time at which the acceleration of the particle is zero, we will follow these steps:

### Step 1: Understand the Position Function
The position \( x \) of the body is given by the equation:
\[ x = Pt^2 - Qt^3 \]
where \( P \) and \( Q \) are constants.

### Step 2: Find the Velocity
To find the acceleration, we first need to find the velocity \( v \) of the particle. The velocity is the first derivative of the position with respect to time \( t \):
\[ v = \frac{dx}{dt} = \frac{d}{dt}(Pt^2 - Qt^3) \]
Using the power rule of differentiation, we get:
\[ v = 2Pt - 3Qt^2 \]

### Step 3: Find the Acceleration
Next, we find the acceleration \( a \) by taking the derivative of the velocity:
\[ a = \frac{dv}{dt} = \frac{d}{dt}(2Pt - 3Qt^2) \]
Again, using the power rule, we find:
\[ a = 2P - 6Qt \]

### Step 4: Set the Acceleration to Zero
To find the time when the acceleration is zero, we set the acceleration equation to zero:
\[ 2P - 6Qt = 0 \]

### Step 5: Solve for Time \( t \)
Rearranging the equation gives:
\[ 6Qt = 2P \]
Now, divide both sides by \( 6Q \):
\[ t = \frac{2P}{6Q} = \frac{P}{3Q} \]

### Conclusion
Thus, the time at which the acceleration of the particle is zero is:
\[ t = \frac{P}{3Q} \]

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