Verify Book Solution for Radius of Curvature Problem

In summary, the conversation was about determining the speed of a projectile launched at an angle of 25 degrees relative to the horizontal, where its radius of curvature is 3/4 of its radius of curvature at the launch point. Both parties independently calculated a speed of 41.6 m/s, but the book's solution was 54.5 ft/s. This was verified using the formula for radius of curvature and the fact that the curvature becomes 3/4 of the initial curvature. The book's solution was found to be correct, regardless of the exact launch angle.
  • #1
skeeter
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1
I was helping someone with the following problem...

A projectile is launched from point A at an angle of 25 degrees relative to the horizontal, with initial velocity = 60 ft/s.
Determine the speed of the projectile along its trajectory where its radius of curvature is 3/4 of its radius of curvature at point A.
(Ignore air resistance.)

We both (independently) worked out a speed = 41.6 m/s, however, the book solution has a speed of 54.5 ft/s.

Appreciate it if someone is able to verify the book's solution.
 
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  • #2
In general, the radius of curvature of a path is given by ##R=\frac{|\mathbf v|^3}{|\mathbf v\times \mathbf{\dot v}|}##
(where ##\mathbf v## is the velocity along the path and ##\mathbf{ \dot v}## is the acceleration)

In the case of a projectile launched at Earth's surface with speed ##v_0## at an angle θ above the horizontal, (and letting time t=0 be the moment of launch) we have:
##\mathbf v = <v_0\cos\theta, v_0\sin\theta-gt, 0>##
##\mathbf{ \dot v} = <0, -g, 0>##
(I include the third, redundant, component just because the cross product is technically only defined in three dimensions.)

We can thus see that the radius of curvature in this case is proportional to the cube of the speed (specifically ##R=\frac{|\mathbf v|^3}{gv_0\cos\theta}##).
This means that if the curvature becomes 3/4 of the
initial curvature, then the speed must become ##\sqrt[3]{3/4}## of the initial speed.

60*(3/4)1/3≈54.5, so the book is indeed correct.


(Notice that we didn't actually need to know the exact launch angle; we only need to know that it wasn't launched straight up.)
 
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What is the radius of curvature?

The radius of curvature is the distance between the center of a circle and its outer edge. In the context of a curve or line, it is the radius of the circle that best fits the curve at a specific point.

Why is it important to verify the book solution for radius of curvature?

Verifying the book solution for radius of curvature is important because it ensures that the calculated value is correct and can be relied upon for further calculations or analysis. It also helps to identify any errors or mistakes in the calculation process.

What factors affect the radius of curvature?

The radius of curvature can be affected by various factors such as the slope of the curve, the size of the curve, and the material properties of the curve. In general, a steeper curve will have a smaller radius of curvature, and a larger curve will have a larger radius of curvature.

How is the radius of curvature calculated?

The radius of curvature can be calculated using the formula: R = [(1 + (dy/dx)^2)^(3/2)] / |d^2y/dx^2|, where dy/dx is the slope of the curve and d^2y/dx^2 is the second derivative of the curve. Alternatively, it can also be calculated using the arc length formula, R = (1 + (dy/dx)^2)^(3/2) / |d^2y/dx^2|, where s is the arc length of the curve.

Are there any limitations to using the book solution for radius of curvature?

There can be limitations to using the book solution for radius of curvature, depending on the complexity of the curve and the accuracy of the given values. It is important to check for any assumptions made in the book solution and make adjustments accordingly for more accurate results.

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