0
$\begingroup$

I need some help with the following question:

A hill can be modeled with the equation $H=100−x^4−3y^2$, where $H$ denotes the elevation.

Now a ball is placed on the hill at position $(x_0,y_0)$. Find the initial direction in which the ball would roll if it would be released from rest. Express this as a cartesian vector.

So far I have: $f(x,y) =100-x^4-3y^2$, $\nabla f(x,y)=-4x^3\hat{i}-6y\hat{j}.$ But I don't know in which direction the ball would roll when $(x_0,y_0)$ and how to express this as a cartesian vector.

$\endgroup$
0
$\begingroup$

The gradient $\nabla f$ gives you the normal to the surface (your hill). To see in which direction the ball would roll, you have to project a downward pointing vector (representing gravitational attraction to the earth) on the plane normal to $\nabla f$ (which corresponds to substracting the normal force applied on the ball by the hill).

$\endgroup$
  • $\begingroup$ Can you please show me how to do this? $\endgroup$ – Jody Sep 26 '14 at 15:22
  • $\begingroup$ Take $v=(0,0,-1)^T$, then the direction in which the ball will start rolling is $v-\frac{v\cdot\nabla f(x_0,y_0)}{\|\nabla f(x_0,y_0)\|}\nabla f(x_0,y_0)$. $\endgroup$ – Daniel Robert-Nicoud Sep 26 '14 at 15:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.