Component of force field tangent to a curve

Question

I recently had to solve the following problem:

A force $F=10\hat x- 8\hat y$ is applied to an object that is constrained to travel towards increasing values of x along the path defined by $y=x^2$ and $z=0$. Find the component of $F$ that is tangent to this path at the point $(2,4,0)$.

Attempt: I used Pythagorean's theorem to $ F=12.81 N$ since the $x$ and $y$ components are given. I just wasn't sure if this is what the questions is asking for, or is there something else I have do do?

Edit. More generally, suppose that one is given a curve and a force field on the plane. How would one go about computing the component of such a force field in the direction tangent to the curve at a given point?

Answer 1

At the specified point, there is a certain tangent line to the path. Also at that point, the vector field $\mathbf F$ has certain components. We want to know the component of $\mathbf F$ in the direction of the tangent line to the path at the point given.

Here's how I would approach this.

Let's restrict to the $x$-$y$ plane. The curve on which the object is traveling can be written as a parameterized curve in 2D as $$ \mathbf x(s) = (x(s), y(s)) = (s, s^2) $$ The unit tangent vector to this path at parameter value $s$ is $$ \mathbf T(s) = \frac{\mathbf x'(s)}{|\mathbf x(s)|} = \frac{(1, 2s)}{\sqrt{1+4s^2}} $$ The tangential component of a vector field $\mathbf F(\mathbf x)$ at a given point along the curve corresponding to parameter value $s$ is therefore $$ F_T(s) = \mathbf F(\mathbf x(s))\cdot\mathbf T(s) $$ I'll let you try to figure out the rest.

Cheers!