- #1
tronter
- 185
- 1
A boy stands at the peak of a hill which slopes downward uniformly at angle [tex] \phi [/tex]. At what angle [tex] \theta [/tex] from the horizontal should he throw a rock so that is has the greatest range.
Ok, so this is a rotation of the normal [tex] x_{1} - x_{2} [/tex] plane right? So we can use the direction cosines [tex] \lambda_{ij} [/tex] to make this problem easier.
So [tex] x'_{1} = x_{1} \cos \phi + x_{2} \cos \left(\frac{\pi}{2} + \phi \right) [/tex] and [tex] x'_{2} = \cos \theta + \cos \phi [/tex].
Are these the right transformations? Is this the right way to set up the problem? Then just apply the equations of projectile motion? This problem seems pretty difficult if I didn't have these tools available. But basically I am using the following:
[tex]A = \begin{bmatrix} \lambda_{11} & \lambda_{12} & \lambda_{13} \\ \lambda_{21} & \lambda_{22} & \lambda_{23} \\ \lambda_{31} & \lambda_{32} & \lambda_{33} \end{bmatrix} [/tex]
Ok, so this is a rotation of the normal [tex] x_{1} - x_{2} [/tex] plane right? So we can use the direction cosines [tex] \lambda_{ij} [/tex] to make this problem easier.
So [tex] x'_{1} = x_{1} \cos \phi + x_{2} \cos \left(\frac{\pi}{2} + \phi \right) [/tex] and [tex] x'_{2} = \cos \theta + \cos \phi [/tex].
Are these the right transformations? Is this the right way to set up the problem? Then just apply the equations of projectile motion? This problem seems pretty difficult if I didn't have these tools available. But basically I am using the following:
[tex]A = \begin{bmatrix} \lambda_{11} & \lambda_{12} & \lambda_{13} \\ \lambda_{21} & \lambda_{22} & \lambda_{23} \\ \lambda_{31} & \lambda_{32} & \lambda_{33} \end{bmatrix} [/tex]
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