A particle travels m a straight line, such that for a short time `2 s le tle 6 s,` its motion is described by `v= (4/a) m//s,` where a is in `m//s^2.` If `v= 6 m//s.` when `t= 2 s,` determine the particle's acceleration when `t= 3 s.`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship between velocity and acceleration
We are given the equation for velocity \( v \) in terms of acceleration \( a \):
\[ v = \frac{4}{a} \]
From this, we can express acceleration in terms of velocity:
\[ a = \frac{4}{v} \]

### Step 2: Use the information given at \( t = 2 \, \text{s} \)
At \( t = 2 \, \text{s} \), we know that \( v = 6 \, \text{m/s} \). We can now find the acceleration at this time:
\[ a = \frac{4}{6} = \frac{2}{3} \, \text{m/s}^2 \]

### Step 3: Relate velocity to time
We can relate the change in velocity to time using the equation:
\[ v \frac{dv}{dt} = 4 \]
This can be rewritten as:
\[ v \, dv = 4 \, dt \]

### Step 4: Integrate to find the velocity as a function of time
We integrate both sides:
\[ \int v \, dv = \int 4 \, dt \]
This gives us:
\[ \frac{v^2}{2} = 4t + C \]
We need to determine the constant \( C \).

### Step 5: Use the initial condition to find \( C \)
At \( t = 2 \, \text{s} \), \( v = 6 \, \text{m/s} \):
\[ \frac{6^2}{2} = 4(2) + C \]
\[ 18 = 8 + C \implies C = 10 \]

### Step 6: Write the expression for velocity as a function of time
Now we can write the equation for velocity:
\[ \frac{v^2}{2} = 4t + 10 \]
Multiplying through by 2 gives:
\[ v^2 = 8t + 20 \]

### Step 7: Find the velocity at \( t = 3 \, \text{s} \)
Substituting \( t = 3 \):
\[ v^2 = 8(3) + 20 = 24 + 20 = 44 \]
Thus,
\[ v = \sqrt{44} = 2\sqrt{11} \, \text{m/s} \]

### Step 8: Find the acceleration at \( t = 3 \, \text{s} \)
Now we can find acceleration using the relationship \( a = \frac{4}{v} \):
\[ a = \frac{4}{\sqrt{44}} = \frac{4}{2\sqrt{11}} = \frac{2}{\sqrt{11}} \, \text{m/s}^2 \]
Calculating this gives approximately:
\[ a \approx 0.603 \, \text{m/s}^2 \]

### Final Answer
The particle's acceleration when \( t = 3 \, \text{s} \) is approximately \( 0.603 \, \text{m/s}^2 \).

---