1. Write the initial and final velocities as vectors

Take east as $+x$ and north as $+y$.

• Initial velocity: $\vec u = 5\hat i \text{ m/s}$
• Final velocity after $10\,$s: $\vec v = 5\hat j \text{ m/s}$

2. Find the change in velocity

Average acceleration is defined as: $\vec a_{\text{avg}} = \frac{\Delta \vec v}{\Delta t} = \frac{\vec v - \vec u}{10}$

So, $\Delta \vec v = 5\hat j - 5\hat i = -5\hat i + 5\hat j$

Hence, $\vec a_{\text{avg}} = \frac{-5\hat i + 5\hat j}{10} = -\frac{1}{2}\hat i + \frac{1}{2}\hat j$

3. Find the magnitude of average acceleration

$|\vec a_{\text{avg}}| = \sqrt{\left(-\frac12\right)^2 + \left(\frac12\right)^2}$ $= \sqrt{\frac14 + \frac14} = \sqrt{\frac12} = \frac{1}{\sqrt2}\text{ m/s}^2$

4. Find the direction

The components are:

• negative in $x$ direction $\Rightarrow$ west
• positive in $y$ direction $\Rightarrow$ north

So the acceleration is towards north-west.

5. Match with the options

• A: $\frac12\text{ m/s}^2$ towards north $\rightarrow$ incorrect
• B: $\frac{1}{\sqrt2}\text{ m/s}^2$ towards north-east $\rightarrow$ incorrect
• C: $\frac{1}{\sqrt2}\text{ m/s}^2$ towards north-west $\rightarrow$ correct
• D: zero $\rightarrow$ incorrect

Therefore, the correct option is C.