A particle is moving along the curve . As the particle passes through the point , its -coordinate increases at a rate of units per second. Find the rate of change of the distance from the particle to the origin at this instant.
To find the rate of change of the distance from the particle to the origin at a specific instant, we need to determine the relationship between the particle's coordinates and its distance from the origin.
Let's assume that the particle's coordinates at any given time are given by (x, y), and the distance from the origin to the particle is given by r.
According to the Pythagorean theorem, the distance from the origin to the particle can be found using the equation:
r = √(x^2 + y^2)
To find the rate of change of the distance from the particle to the origin, we need to differentiate both sides of this equation with respect to time.
d(r)/dt = d(√(x^2 + y^2))/dt
Applying the chain rule, we get:
d(r)/dt = (1/2)*(x^2 + y^2)^(-1/2) * d(x^2 + y^2)/dt
Now, let's analyze the given information. We know that the particle's y-coordinate increases at a rate of units per second. In other words:
dy/dt = units/sec
And since we are interested in finding the rate of change of the distance from the particle to the origin when it passes through a specific point, we can assume that (x, y) represents the coordinates of that point. Therefore, we have:
x = x (constant)
y = y (value changes as the particle moves)
Now, let's differentiate the equation for r with respect to time:
d(r)/dt = (1/2)*(x^2 + y^2)^(-1/2) * [d(x^2)/dt + d(y^2)/dt]
Since x is constant, its derivative is zero:
d(x^2)/dt = 0
Next, let's consider the given information: "As the particle passes through the point , its y-coordinate increases at a rate of units per second." This means that at the specific point, we have:
dy/dt = units/sec
Therefore:
d(y^2)/dt = 2y * (dy/dt)
Substituting the values we derived:
d(r)/dt = (1/2)*(x^2 + y^2)^(-1/2) * (0 + 2y * (dy/dt))
Simplifying further:
d(r)/dt = y * (dy/dt) / √(x^2 + y^2)
This is the expression for the rate of change of the distance from the particle to the origin at the particular instant.