| Problem statement Solution video |
DISCUSSION THREAD
Discussion and hints:
For this problem, the “Given” information is in terms of the Cartesian kinematical description. The “Find” asks for parameters that are part of the path kinematical description.
For the Given part of the kinematics, you will need to find the first and second derivatives of y with respect to time using implicit differentiation of the the path equation: xy = b, and using that the first time derivative of x is a constant value of c.
Next, you need to relate the Cartesian and path descriptions of the acceleration. There are many approaches in doing this. The most straight-forward approach is to use the projection of acceleration onto the unit tangent vector to find the rate of change of speed: v_dot = aP • et, where the unit tangent is found from et = vP/vP. The radius of curvature can then be found directly from the magnitude of acceleration equation: aP2 = v_dot2 + (vP2/ρ )2.
Recall that the sign of v_dot tells us whether P is increasing or decreasing in speed. From the animation below, we see that the acceleration vector has a negative et throughout the full range of motion. Alternately, you can say that the angle between velocity and acceleration is greater than 90° over that motion. So: increasing or decreasing in speed?
Any questions??
Should we be searching for the Theta between the different cartesian unit vectors and the path based ones? Or should this be able to be down purely through vector projection?
I assume that you were able to get through Part (a) of finding v and a in terms of their Cartesian components.
Then in Part (b) you can find the e_t unit vector in terms of its Cartesian components directly from v (simply divide by the speed).
In Part (d) use a projection to find v_dot using a•e_t. The dot product is possible simply through the operation of the dot products of two vectors written in terms of Cartesian components. There is no need to find any angles.
As suggested above, you can find rho through the magnitude of the acceleration equation.
In short, you can avoid finding angles.
I didn’t quite understand the reasoning behind using dot products for this problem.
You are asked to find the rate of change of speed (v_dot) of P. v_dot is the tangential component of the acceleration. To find this tangential component, you will want to project the acceleration vector a onto the tangential unit vector. The most direct way to perform a projection is through a dot product.
As O asks above, you could find the projection through trig; however, that would involve first finding the angle between a and e_t. The dot product is a simpler way to find the projection.