Diff Eqs for a block moving sideways down a plane

[joej24]

Homework Statement

I was looking at the problem discussed in the thread below.
https://www.physicsforums.com/showthread.php?t=502417

A block is initially kicked in the direction perpendicular to the downslope of the plane with initial speed V.
The problem asks for the speed of the block down the plane after a long time if the coeff of friction is tan \theta.

Homework Equations

The thread I posted above mentions using the fact that \sqrt{v_x^2 + v_y^2} + v_y = C, where C is a constant, since the speed gained in the y-direction will be equal to the speed lost overall.

C is initially V. Eventually v_x goes to zero and v_y goes to v_f. So
v_f + v_f = V and the speed down the plane after a long period of time is \frac{V}{2}

However, I am interested if this problem can be solved in another way, using the equations of motion.

The Attempt at a Solution

If \gamma = arctan \frac{v_y}{v_x}, then the equations of motion for the x (perpendicular to the downslope) and y (down the slope) directions respectively will be

-mg\,sin\theta\, cos \gamma = m \ddot x
mg\,sin\theta - mg\,sin\theta\, sin\gamma = m \ddot y

These simplify to

\frac {-gsin\theta}{\sqrt{\frac{v_y^2}{v_x^2} + 1}} = \ddot x
gsin\theta \,(1-\frac{\frac{v_y}{v_x}}{\sqrt{\frac{v_y^2}{v_x^2} +1 }}) = \ddot y

Also, v_y = \dot y and v_x = \dot x. The total speed, V = v_y^2 + v_x^2, is constant too. So we further can further simplify the equations to these:

\frac{-gsin\theta}{V} \dot x = \ddot x
gsin\theta \,(1-\frac{\dot y}{V}) = \ddot y

I am wondering how to proceed in solving these equations. I am trying to find \dot y.

 

[SteamKing]

\frac{-gsin\theta}{V} \dot x = \ddot x
gsin\theta \,(1-\frac{\dot y}{V}) = \ddot y

I am wondering how to proceed in solving these equations. I am trying to find \dot y.

You can't take the next step algebraically and solve for \dot y?

Also, shouldn't The total speed be v^2 = v^2_y+v^2_x?

 

[joej24]

Right, the total speed should be v^2, not v. Then the square root makes it v.

Seperating the equation for \dot y,
\dot y \, = \, v \, (1 \, - \, \frac{\ddot y}{g\,sin\theta}).

As t -> \infty, \ddot y -> 0 because the block will be moving straight down the incline. This can't be correct though, because that would mean \dot y -> v.

Finding y(t) isn't as hard as I thought. I can integrate both sides of the motion equation to obtain v \,( t - \frac{\dot y}{g\,sin\theta}) = y + y_0. But we can say that y(0) = 0 to get rid of the y_0 term.

Then I rearrange the terms to get \dot y + (\frac {g\,sin\theta}{v}) \, y = (g\,sin\theta)\,t.
This is a first order linear diff eq that's solvable.

y = Ce^{-\frac{g\,sin\theta}{v} t} + e^{-\frac{g\,sin\theta}{v} t} \int (g\,sin\theta) \,t \, e^{\frac{g\,sin\theta}{v} t}

I simplified this down to
y = vt - \frac{v^2}{g\,sin\theta} + Ce^{\frac{-g\,sin\theta}{v}t}

So,

\dot y = v - \frac{g\,sin\theta}{v} C e^{\frac{-g\,sin\theta}{v}t}.

But, this also suggests that \dot y goes to v instead of \frac{v}{2}.

 

[ehild]

Homework Statement

I was looking at the problem discussed in the thread below.
https://www.physicsforums.com/showthread.php?t=502417

A block is initially kicked in the direction perpendicular to the downslope of the plane with initial speed V.
The problem asks for the speed of the block down the plane after a long time if the coeff of friction is tan \theta.

2.

However, I am interested if this problem can be solved in another way, using the equations of motion.

The Attempt at a Solution

If \gamma = arctan \frac{v_y}{v_x}, then the equations of motion for the x (perpendicular to the downslope) and y (down the slope) directions respectively will be

-mg\,sin\theta\, cos \gamma = m \ddot x
mg\,sin\theta - mg\,sin\theta\, sin\gamma = m \ddot y

These simplify to

\frac {-gsin\theta}{\sqrt{\frac{v_y^2}{v_x^2} + 1}} = \ddot x
gsin\theta \,(1-\frac{\frac{v_y}{v_x}}{\sqrt{\frac{v_y^2}{v_x^2} +1 }}) = \ddot y

Also, v_y = \dot y and v_x = \dot x. The total speed, V = v_y^2 + v_x^2, is constant too.

Why should be the total speed constant?



ehild

 

[dauto]

I think the whole premiss that C as defined at the OP is a constant is just pain wrong. Why would that be the case?

 

[ehild]

See https://www.physicsforums.com/showthread.php?t=753170&highlight=utkarshakash&page=2 Post#20

If v is the speed and v_y is the velocity component down the incline, it comes out that dv/dt = -dv_y/dt. That means v+v_y=const.

ehild

 

[joej24]

Why should be the total speed constant?
ehild

That's a mistake. I realize now that total speed + v_y is a constant, \sqrt{v_x^2 + v_y^2} + v_y = C. I will check out the link you posted, which is the same problem, and try to redo my math.