Question

5) Placido pulls a rope attached to a wagon through a pulley at a rate of q m/s. With dimension as in Figure 5:

Answer 1

The formula for the speed of the wagon can be found by using Pythagoras theorem, and chain rule of differentiation.

$(a) \ \mathrm{The \ speed \ of \ the \ wagon} , \ \dfrac{dx}{dt} = q -\dfrac{x}{\sqrt{x^2+ 5.76}}$

(b) When x = 0.6 and q = 0.5 m/s, the speed of the wagon is approximately 0.243 m/s.

Reasons:

(a) The distance from the cart to the pulley, r is given by Pythagoras's

theorem as follows;

r = √((3 - 0.6)² + x²) = √(2.4² + x²)

The speed of the wagon = $\mathbf{\dfrac{dx}{dt}}$

The speed of the rope, q = $\mathbf{\dfrac{dr}{dt}}$

By chain rule, we have;

$\dfrac{dr}{dt} = \mathbf{\dfrac{dr}{dx} + \dfrac{dx}{dt}}$

$\dfrac{dr}{dx} = \dfrac{x}{\sqrt{x^2+ 5.76}}$

Therefore;

$\dfrac{dr}{dt} = q =\dfrac{x}{\sqrt{x^2+ 5.76}}+ \dfrac{dx}{dt}$

$\mathrm{The \ speed \ of \ the \ wagon} , \ \dfrac{dx}{dt} = \underline{ q -\dfrac{x}{\sqrt{x^2+ 5.76}}}$

(b) The speed of the wagon when x = 0.6 if q = 0.5 m/s is given as follows;

$\dfrac{dx}{dt} = 0.5 -\dfrac{0.6}{\sqrt{0.6^2+ 5.76}} =\sqrt{\dfrac{1}{17} } \approx 0.243$

The speed of the wagon when x = 0.6 and q = 0.5 m/s, $\dfrac{dx}{dt}$ ≈ 0.243 m/s

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