The position vector of a particle is given by `vecr=vecr_(0)` (1-at)t, where t is the time and a as well as `vecr_(0)` are constant. After what time the particle retursn to the starting point?

Options

A. a
B. `(1)/(a)`
C. `a^(2)`
D. `(1)/(a^(2))`

Explanation

To solve the problem step by step, we start with the given position vector of the particle:

### Step 1: Write down the position vector
The position vector of the particle is given by:
\[ \vec{r} = \vec{r_0} (1 - a t) t \]
where \( \vec{r_0} \) and \( a \) are constants, and \( t \) is the time.

### Step 2: Determine the condition for returning to the starting point
The particle returns to the starting point when the position vector \( \vec{r} \) becomes zero. Therefore, we set:
\[ \vec{r} = 0 \]

### Step 3: Set the equation to zero
From the position vector equation, we have:
\[ \vec{r_0} (1 - a t) t = 0 \]

### Step 4: Analyze the equation
This equation can be satisfied in two ways:
1. \( t = 0 \) (the initial time)
2. \( (1 - a t) = 0 \)

### Step 5: Solve for time \( t \)
From the second condition:
\[ 1 - a t = 0 \]
This implies:
\[ a t = 1 \quad \Rightarrow \quad t = \frac{1}{a} \]

### Step 6: Conclusion
Thus, the particle returns to the starting point at:
\[ t = \frac{1}{a} \]

### Final Answer
The time after which the particle returns to the starting point is:
\[ t = \frac{1}{a} \]

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