# What direction will the ball start to roll?

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.

## 1 Answer

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).

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