Given:
Particles P and Q are initially 40m apart, with P behind Q.
Particle P has a uniform velocity of 10m/s towards Q.
Particle Q starts from rest and has an acceleration of 2m/s^2 in the direction of P's velocity.

To find:
The minimum distance between particles P and Q.

Explanation:
Let's analyze the motion of particles P and Q to find their minimum distance.

1. Position of Particle P:
Since particle P has a uniform velocity of 10m/s towards Q, its position can be described by the equation:
xP = x0 + vt
Where xP is the position of P at time t, x0 is the initial position of P, v is the velocity of P, and t is the time.

2. Position of Particle Q:
Particle Q starts from rest and has an acceleration of 2m/s^2 in the direction of P's velocity.
Using the equations of motion, we can find the position of Q at time t:
xQ = x0 + ut + (1/2)at^2
Where xQ is the position of Q at time t, x0 is the initial position of Q, u is the initial velocity of Q (which is 0 in this case), a is the acceleration of Q, and t is the time.

3. Time of Collision:
To find the minimum distance between P and Q, we need to determine the time at which they collide.
Since both particles start at the same time, their time of collision can be found by setting their positions equal to each other:
xP = xQ
x0P + vPt = x0Q + uQt + (1/2)aQt^2
Since x0P = 40m (initial position of P) and x0Q = 0m (initial position of Q), the equation becomes:
40 + 10t = 0 + 0 + (1/2)(2)(t^2)
Simplifying the equation:
10t = t^2
Solving for t:
t^2 - 10t = 0
t(t - 10) = 0
t = 0 or t = 10

Since t cannot be 0 (as both particles start moving simultaneously), the time of collision is t = 10s.

4. Minimum Distance:
To find the minimum distance between P and Q, we substitute the time of collision (t = 10s) into the position equations for P and Q:
xP = x0P + vPt
xP = 40 + (10)(10)
xP = 40 + 100
xP = 140m

xQ = x0Q + uQt + (1/2)aQt^2
xQ = 0 + 0 + (1/2)(2)(10^2)
xQ = 0 + 0 + (1/2)(2)(100)
xQ = 0 + 0 + 100
xQ = 100m

The minimum distance between P and Q is given by the difference in their positions:
Minimum distance = xP - xQ
Minimum distance = 140 - 100
Minimum distance = 40