A particle of mass m moves on a straight line with its velocity varying with the distance travelled according to the equation `v=asqrtx`, wher ea 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, we need to find the total work done by all the forces during the displacement from \( x = 0 \) to \( x = d \). We will use the work-energy theorem, which states that the total work done on an object is equal to the change in its kinetic energy.

### Step-by-Step Solution:

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

2. **Determine the initial velocity**:
At \( x = 0 \):
\[ v(0) = a \sqrt{0} = 0 \]
Therefore, the initial velocity \( v_i = 0 \).

3. **Determine the final velocity**:
At \( x = d \):
\[ v(d) = a \sqrt{d} \]
Therefore, the final velocity \( v_f = a \sqrt{d} \).

4. **Calculate the initial kinetic energy**:
The initial kinetic energy \( KE_i \) at \( x = 0 \) is given by:
\[ KE_i = \frac{1}{2} m v_i^2 = \frac{1}{2} m (0)^2 = 0 \]

5. **Calculate the final kinetic energy**:
The final kinetic energy \( KE_f \) at \( x = d \) is given by:
\[ KE_f = \frac{1}{2} m v_f^2 = \frac{1}{2} m (a \sqrt{d})^2 = \frac{1}{2} m a^2 d \]

6. **Calculate the change in kinetic energy**:
The change in kinetic energy \( \Delta KE \) is:
\[ \Delta KE = KE_f - KE_i = \frac{1}{2} m a^2 d - 0 = \frac{1}{2} m a^2 d \]

7. **Total work done**:
According to the work-energy theorem, the total work done \( W \) is equal to the change in kinetic energy:
\[ W = \Delta KE = \frac{1}{2} m a^2 d \]

### Final Answer:
The total work done by all the forces during the displacement from \( x = 0 \) to \( x = d \) is:
\[ W = \frac{1}{2} m a^2 d \]