To determine how rapidly the angle between the x-axis and the line joining the two particles is changing, we can use trigonometry.
Let's consider the two particles as A and B, with coordinates (5,0) and (0,7) respectively.
The angle between the x-axis and the line joining A and B can be found using the tangent function. The tangent of an angle is defined as the ratio of the opposite side to the adjacent side. In this case, the opposite side is the y-coordinate difference between A and B (Δy = 7 - 0 = 7), and the adjacent side is the x-coordinate difference between A and B (Δx = 5 - 0 = 5).
So, the tangent of the angle θ (theta) between the x-axis and the line joining A and B is given by: tan(θ) = Δy / Δx = 7 / 5.
To find how rapidly the angle is changing, we need to differentiate this expression with respect to time (t). Since both Δy and Δx are changing with time, we can write:
tan(θ) = Δy / Δx
Differentiating both sides with respect to t using implicit differentiation:
sec²(θ) * dθ/dt = (dΔy/dt * Δx - Δy * dΔx/dt) / Δx²
We already know that dΔy/dt = -3 and dΔx/dt = 2. Substituting these values into the equation:
sec²(θ) * dθ/dt = (-3 * 5 - 7 * 2) / (5²) = (-15 - 14) / 25 = -29 / 25
Now, we can solve for dθ/dt, the rate at which the angle is changing:
dθ/dt = (-29 / 25) * (1 / sec²(θ))
Since sec²(θ) is the reciprocal of cos²(θ), we can rewrite this as:
dθ/dt = (-29 / 25) * (1 / (1 + tan²(θ)))
Now, we substitute the value of tan(θ) that we found previously:
dθ/dt = (-29 / 25) * (1 / (1 + (7 / 5)²))
Simplifying further:
dθ/dt = (-29 / 25) * (1 / (1 + 49 / 25))
= (-29 / 25) * (1 / (74 / 25))
= -29 / 74
Therefore, the rate at which the angle between the x-axis and the line joining the two particles is changing at that instant is -29/74 units per second.
To determine if the particles are moving towards or away from each other at that instant, we need to check the relative velocity vector between the particles.
The relative velocity vector is given by subtracting the velocity vector of particle B from the velocity vector of particle A. The velocity vector of A is (2,0) and the velocity vector of B is (0,-3).
Relative velocity vector = Velocity vector of A - Velocity vector of B
= (2,0) - (0,-3)
= (2,3)
Therefore, the particles are moving towards each other at that instant, as the relative velocity vector has both x and y components positive.