Transformations physics problem

In summary, the problem involves finding the optimal angle from the horizontal at which a boy should throw a rock from the peak of a hill with a downward slope of angle phi. This can be solved using transformation matrices and the equations of projectile motion, but the suggested transformations and use of a three-dimensional transform may not be the most efficient approach. Additionally, the commonly held belief that 45 degrees is always the most effective angle is not necessarily true in this case.
  • #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]
 
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  • #2
anybody?
 
  • #3
You don't really need to do that. The primary reason for transforming frames is to make the problem easier to solve. This transformation makes the problem harder to solve, not easier. And why a three-dimensional transform? This is a 2D problem.
 
  • #4
At the risk of sounding like an uneducated fool:

Isn't 45° always the most effective angle to use?
 
  • #5
sephirothrr said:
Isn't 45° always the most effective angle to use?

No. At this point in time it would not be appropriate to derive the answer. Perhaps the original poster will. If he/she doesn't, I'll be glad to do so after a few days. For now, this is a homework thread.
 

What is a transformation physics problem?

A transformation physics problem is a type of problem in which the state of an object or system is changed or transformed in some way, and the goal is to determine the final state or the changes that occur during the transformation.

What types of transformations are commonly encountered in physics problems?

Common types of transformations in physics problems include translations (movement in space), rotations (spinning), and accelerations (change in velocity). Other types of transformations include changes in temperature, pressure, and electric or magnetic fields.

How are transformations represented in physics problems?

Transformations are often represented using mathematical equations or diagrams, such as vector diagrams or coordinate systems. These representations help to visualize and understand the changes that occur during the transformation.

What are some real-life examples of transformation physics problems?

Real-life examples of transformation physics problems include calculating the trajectory of a rocket during launch, determining the final velocity of a car after braking, and calculating the change in temperature of a cup of coffee as it cools down.

What skills are required to solve transformation physics problems?

Solving transformation physics problems requires a strong understanding of fundamental physics concepts, such as motion, forces, and energy. It also requires the ability to use mathematical equations and analyze data to make predictions and solve problems.

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