Answer :
Answer:
(dθ/dt) = 1 rad / s = 57.3° /s
Step-by-step explanation:
- A particle moves along curve with function:
y = x^2
- The rate of change of x-coordinate is given by dx/dt = 10 m/s
Find:
How fast is the angle of inclination theta of the line joining the particle to the origin changing when x equals 1 m
Solution:
- The gradient of the line from origin to particle at position ( x , y ) is given by:
tan ( θ ) = y / x
Where, θ is the angle between x-axis and line from origin
x & y are coordinate of the point on given graph.
- To develop a rate of change expression we will derivate the above expression by time t:
d / dt (tan ( θ )) = d/dt (y / x )
(dθ/dt) / cos^2(θ) = (dy/dt) / (dx/dt)
(dθ/dt) = cos^2(θ) * (dy/dt) / (dx/dt)
- The rate of change of angle (dθ/dt) is given by above expression.
- We will apply the following chain rule to evaluate (dy/dt):
(dy/dt) = (dy/dx) * (dx/dt)
(dy/dt) = 2x * (10)
(dy/dt) = 20*x
@ x = 1, (dy/dt) = 20 m/s
@ x = 1, y = (1)^2 = 1
tan (θ) = 1
θ = 45°
- Now use the derived rate of change of angle expression we get:
(dθ/dt) = cos^2(45) * 20 / 10
(dθ/dt) = 0.5 * 20 / 10
(dθ/dt) = 1 rad / s = 57.3° /s