A particle of mass m moves on a straight line with its velocity varying with the distance travelled according to the equation `v=asqrtx`, where a is a constant. Find the total work done by all the forces during a displacement from `x=0 to x=d`.

Explanation

To solve the problem of finding the total work done by all forces during the displacement of a particle from \( x = 0 \) to \( x = d \), we can use the work-energy theorem. This theorem states that the work done by all forces acting on a particle is equal to the change in its kinetic energy.

### Step-by-Step Solution:

1. **Identify the given equation for velocity**:
The velocity of the particle is given by the equation:
\[ v = a \sqrt{x} \]
where \( a \) is a constant.

2. **Determine the initial and final velocities**:
- At \( x = 0 \):
\[ v_{\text{initial}} = a \sqrt{0} = 0 \]
- At \( x = d \):
\[ v_{\text{final}} = a \sqrt{d} \]

3. **Calculate the change in kinetic energy**:
The kinetic energy (\( KE \)) of an object is given by:
\[ KE = \frac{1}{2} mv^2 \]
- Initial kinetic energy at \( x = 0 \):
\[ KE_{\text{initial}} = \frac{1}{2} m (0)^2 = 0 \]
- Final kinetic energy at \( x = d \):
\[ KE_{\text{final}} = \frac{1}{2} m (a \sqrt{d})^2 = \frac{1}{2} m a^2 d \]

4. **Calculate the total work done**:
According to the work-energy theorem:
\[ W = KE_{\text{final}} - KE_{\text{initial}} \]
Substituting the values we calculated:
\[ W = \frac{1}{2} m a^2 d - 0 = \frac{1}{2} m a^2 d \]

Thus, the total work done by all the forces during the displacement from \( x = 0 \) to \( x = d \) is:
\[ \boxed{\frac{1}{2} m a^2 d} \]