- #1

danyull

- 9

- 1

## Homework Statement

A ball is thrown with initial speed v

_{0}up an inclined plane. The inclined plane makes an angle of π/6 above the horizontal line and the ball is launched at an angle θ above the inclined plane. No air resistance in this problem.

(a) How long does the ball stay in the air?

(b) At what angle θ should the ball be launched in order to fall back on the plane normal to the inclined surface?

## Homework Equations

Newton's laws and kinematics equations?

## The Attempt at a Solution

I tried making a rotated reference frame (expressed as a matrix transformation below) centered at the launch point where $$ \begin{pmatrix} x' \\ y'

\end{pmatrix} = \begin{pmatrix} \cos(π/6) & \sin(π/6) \\ -\sin(π/6) & \cos(π/6) \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} \\

\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} \frac {\sqrt{3}} 2 & \frac 1 2 \\ - \frac 1 2 & \frac {\sqrt{3}} 2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}

$$

and since the force of gravity is ##\vec {\mathbf F} = -mg ~ \hat y##, then in the new reference frame $$ F_x' = \frac {\sqrt{3}} 2 F_x + \frac 1 2 F_y = - \frac 1 2 mg \\ F_y' = - \frac 1 2 F_x + \frac {\sqrt{3}} 2 F_y = - \frac {\sqrt{3}} 2 mg$$

and acceleration ##a_x'## and ##a_y'## will just be the above two forces with mass divided out.

From here on I'm a bit stuck. As for part (a), I want to use the kinematic equations with my given initial velocity. At first I thought to solve for the time it takes to reach the peak of its trajectory and then multiplying by 2, but I realized that would be the time it takes to hit the ground if the inclined plane weren't there. My next guess was solving for the time when the horizontal velocity reached zero since in this reference frame there's a negative horizontal acceleration, but I'm not sure that would be the exact time that the ball hits the inclined plane.

Any help would be appreciated!