Newtonian Mechanics – Direction and Magnitude of Kinetic Friction for Body Sliding Up Inclined Plane

Question

So I have a body of mass $m$ at the bottom of the inclined plane, and I project it up the incline with velocity $u$. I want to find the time it will take to reach the maximum height up the incline. I know the final velocity $v$ at this point will be $0$. The coefficient of friction between the surfaces is $\mu$ and the angle of incline is greater than the angle of repose, so kinetic friction is $\mu mg \cos \theta$, where $\theta$ is the angle of incline of the plane. Since velocity of the block would be up the incline, friction should act down the incline. So the acceleration on the mass down the incline would be $g \sin\theta + \mu \cos \theta$, right? But my teacher has written $g \sin\theta – \mu \cos \theta$, (down the incline) which implies that friction is pushing the body further up the incline, which doesn't make sense to me. Is this wrong, or am I missing something?

(In fact, I think this would be the correct acceleration if the body was sliding down the incline.)

Answer 1

Use those rules

for a given velocity direction $~v~$

• the inertia force $~m\dot v~$ is opposite to the velocity direction
• the friction force $~F_\mu~$ is also opposite to the velocity direction
• obtain the sum of all forces equal zero $~\sum F_i=0~$ and solve for the acceleration $~\dot v~$