x=6+7t+18tsqareNow, velocity of the partocle-V=dx/dt=7+36t.....(i)Now comparing the equation (i) with the equation [v=u+at] we get u=7.. so the initial velocity of the particle is 7m/s

**Finding the Initial Velocity with Uniform Acceleration**

To find the initial velocity of a particle moving along a straight line path with uniform acceleration, we need to analyze the given equation for displacement. The equation for displacement is given as x = 6 + 7t + 18t², where x represents the distance covered by the particle and t represents time.

**Understanding the Equation**

The equation x = 6 + 7t + 18t² represents a quadratic function, where the coefficient of t² is positive. This indicates that the particle is moving along a straight line path with uniform acceleration. The term 6 in the equation represents the initial position of the particle.

**Deriving the Velocity Equation**

To find the initial velocity, we need to derive the velocity equation from the given displacement equation. Velocity is the rate of change of displacement with respect to time. In other words, velocity is the derivative of displacement with respect to time.

Taking the derivative of the displacement equation, we get:

v = dx/dt = d(6 + 7t + 18t²)/dt = 7 + 36t

The velocity equation v = 7 + 36t represents the instantaneous velocity of the particle at any given time t.

**Finding the Initial Velocity**

To find the initial velocity, we substitute t = 0 into the velocity equation. This is because the initial velocity represents the velocity of the particle at t = 0.

v = 7 + 36(0) = 7

Therefore, the initial velocity of the particle is 7 m/s.

**Explanation**

The initial velocity represents the velocity of the particle at the start of its motion (t = 0). By differentiating the displacement equation with respect to time, we obtain the velocity equation. Substituting t = 0 into the velocity equation gives us the initial velocity. In this case, the initial velocity is 7 m/s.