Homework Hint: Problem 2/133

Note that the "Given" information for this problem is in terms of Cartesian components whereas the "Find" information is in terms of path components.

You first need to express velocity and acceleration of P in terms of Cartesian components.

• Determine the equation y = y(x) for the path. You know that the path is parabolic: y(x) = a + b*x^2. Use the coordinates of the two known points to find a and b.
• Use the given x-components of velocity and acceleration of P to find the y-components of velocity v and acceleration a.
• You now have the x-and y-components of v and a.

Now you need to convert from Cartesian components to path components of acceleration. There are a number of ways to do this. One method is:

• Find the Cartesian components of the tangential unit vector e_t using: e_t = v/v, where v is the speed of P.
• Find the rate of change of speed, dv/dt, by projecting a onto e_t: dv/dt = a • e_t (here "•" is the dot product, not a multiplication).
• Find the centripetal component of acceleration using the magnitude of a and the known rate of change of speed: a^2 = (dv/dt)^2 + (v^2/rho)^2. From this, you can solve for the radius of curvature, rho.

Other methods are more graphical. For example, you can make a sketch of v and a. From this find the angle between v and a, from which you can find the projection of a onto the direction perpendicular to v, giving you the centripetal component, v^2/rho. The vector projection method above is recommended.

The problem statement references an equation in Appendix C/10 for the radius of curvature of a path given in terms of its Cartesian components, y = y(x). This is the second-to-the-last equation on page 700.