To solve this problem, we need to analyze the motion of the two particles, P and Q, and how their velocities interact over time. Let's break it down step by step.
Understanding the Motion of the Particles
We have two particles: P, which is moving towards Q, and Q, which is moving horizontally. Initially, they are separated by a distance d. The velocity of particle Q, denoted as u, is constant and directed horizontally, while the velocity of particle P, denoted as v, is directed towards Q. At the start, the motion of Q is perpendicular to that of P.
Setting Up the Problem
To visualize this, imagine a coordinate system where particle Q starts at the origin (0, 0) and moves along the x-axis. Particle P starts at point (0, d) and moves towards Q. As time progresses, the positions of the particles change:
- Position of Q after time t: (ut, 0)
- Position of P after time t: (0, d - vt)
Finding the Meeting Point
For the two particles to meet, their coordinates must be equal. This means we need to set the x-coordinate of Q equal to the x-coordinate of P, and the y-coordinate of P must equal zero (since they meet at the same horizontal level).
From the position of Q, we have:
ut = x-coordinate of Q
From the position of P, we can find the distance to Q at any time t:
Distance = √[(ut - 0)² + (0 - (d - vt))²] = √[u²t² + (d - vt)²]
For the particles to meet, this distance must equal zero:
√[u²t² + (d - vt)²] = 0
Setting Up the Equation
Squaring both sides gives us:
u²t² + (d - vt)² = 0
Expanding the second term:
u²t² + (d² - 2dvt + v²t²) = 0
Combining like terms results in:
(u² + v²)t² - 2dvt + d² = 0
Applying the Quadratic Formula
This is a quadratic equation in the form of at² + bt + c = 0, where:
- a = u² + v²
- b = -2dv
- c = d²
Using the quadratic formula, t can be found as:
t = [2dv ± √((-2dv)² - 4(u² + v²)d²)] / 2(u² + v²)
To find the time when they meet, we focus on the positive root since time cannot be negative. Simplifying this expression leads us to the time of meeting:
Final Result
After simplification, we find that the time t when particles P and Q meet is given by:
t = vd / (v² - u²)
This result shows how the relative velocities and the initial distance between the particles influence the time it takes for them to meet. The key takeaway is that as long as P is moving directly towards Q and Q maintains a constant horizontal velocity, we can predict their meeting time using this formula.
