To find the value of K when particles A and B are closest to each other, we first need to establish their positions as functions of time. Let's break down the problem step by step.
Initial Positions and Velocities
Particle A starts at the point (0, -10√3) and moves with a velocity of 5i m/s. This means it moves in the positive x-direction. The position of particle A at time t can be expressed as:
- Position of A: (5t, -10√3)
Particle B starts at the point (0, 0) and moves with a velocity of -5√3j m/s, indicating it moves in the negative y-direction. The position of particle B at time t is given by:
- Position of B: (0, -5√3t)
Distance Between the Particles
The distance D between the two particles at any time t can be calculated using the distance formula:
- Distance D: D = √[(x2 - x1)² + (y2 - y1)²]
Substituting the positions of A and B into the distance formula, we have:
- D = √[(5t - 0)² + (-10√3 - (-5√3t))²]
This simplifies to:
- D = √[(5t)² + (-10√3 + 5√3t)²]
Expanding the Distance Formula
Now, let's expand the expression:
- D = √[25t² + (5√3t - 10√3)²]
Expanding the squared term:
- (5√3t - 10√3)² = 75t² - 100√3t + 100
Combining these, we get:
- D² = 25t² + 75t² - 100√3t + 100
Thus, the squared distance becomes:
- D² = 100t² - 100√3t + 100
Minimizing the Distance
To find when the particles are closest, we need to minimize D². This can be done by taking the derivative of D² with respect to t and setting it to zero:
- d(D²)/dt = 200t - 100√3 = 0
Solving for t gives:
Finding K
According to the problem, the time when they are closest is given as 3/K seconds. Setting these equal gives:
Cross-multiplying to solve for K:
Final Result
Thus, the value of K is:
This means that the particles A and B are closest to each other at a time of 3/(2√3) seconds. If you have any further questions or need clarification on any part of this process, feel free to ask!
