Answer to Question #128860 in Mechanics | Relativity for Ariyo Emmanuel

Question #128860
When x = 10 ft, the crate has a speed of 20 ft/s which is increasing at 6 ft/s2. Determine the direction of the crate’s velocity and the magnitude of the crate’s acceleration at this instant.
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Expert's answer
2020-08-27T10:25:37-0400

To answer this question we should know the equation of the trajectory of the crate.

Let us consider the trajectory given in this problem as an example: https://www.chegg.com/homework-help/questions-and-answers/3-x-10-ft-crate-speed-20-ft-s-increasing-6-ft-s-2-determine-direction-crate-s-velocity-mag-q8097639


The velocity is directed along the tangent line to the trajectory. Let us calculate the coordinates of the point:   x0=10ft,    y0=124x2=10024=256ft.\,\, x_0=10\,\text{ft},\;\; y_0 = \dfrac{1}{24}x^2 = \dfrac{100}{24} = \dfrac{25}{6}\,\text{ft}.

The tangent line is tilted at an angle with tanα=y(x0)=112100.833,  α39.8.\tan \alpha = y'(x_0) = \dfrac{1}{12}\cdot10 \approx 0.833, \; \alpha \approx 39.8^\circ. The direction of the velocity vector makes an angle 39.8-39.8^\circ with the x-axis.


Let us determine the curvature of the trajectory in point in question:

κ=f(1+f2)3=112(1+2536)3=125126130.0378,\kappa = \dfrac{|f''|}{(\sqrt{1+f'^2})^3} = \dfrac{\dfrac{1}{12}}{\left(\sqrt{1+\dfrac{25}{36}}\right)^3} = \dfrac{125}{12\sqrt{61^3}} \approx 0.0378,

so the radius of curvature is r=1/κ26.5ft.r=1/\kappa \approx 26.5\,\text{ft}.

We know the tangential component of the acceleration (6 ft/s^2) and we should calculate the normal component that is equal to

an=v2r=40026.5=15ft/s2.a_n = \dfrac{v^2}{r}= \dfrac{400}{26.5} = 15\,\text{ft/s}^2.

The total acceleration can be obtained by means of Pythagorean theorem: a=at2+an2=62+152=16.2ft/s2.a= \sqrt{a_t^2+a_n^2} = \sqrt{6^2+15^2} = 16.2\,\text{ft/s}^2.


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