Kinematics problem from a competition: Will the 2 sliding boxes collide?

[Jim Alexandridis]

Homework Statement

two small boxes of rectangular parallelepiped shape are launched simultaneously against each other, with velocities of meters 𝑢1=𝑢 and 𝑢2=2𝑢, from positions A and B of a straight and horizontal trajectory 𝑂𝑥 lying in a plane. The distance between them is 𝛢𝛣=𝑠 and the coefficient of friction between each body and the surface is μ. Boxes will collide if:
a)u≥2√(μgs)/3
b)u≥1√(μgs)/3
c)u≥2√(μgs)/5
d)u≥3√(μgs)/5

This problem was on an greek physics competition on 10th grade and there is a disagreement about the answer, so i would like your opinion depending on my answer

Relevant Equations

Kinematics:
V=v0+at
Dx=v0t+1/2at²
Smax=v²/2a

The distance covered by the first box is :s1max=v²/2|a|=v²/2μg where a=-μg by second newtons law
Similarly S2max=(2v)²/2|a|=4v²/2μg
It gas to be s1max+s2max≥S => v²/2a +4v²/2a ≥s => 5v²≥2aS =>v²≥ 2μgS/5=> v≥√(2μgs/5)
But this is in the possible solution, am I wrong somewhere? I appreciate your help

 

[kuruman]

Hi @Jim Alexandridis and welcome to PF.

The first box travels distance $s_1=\dfrac{u^2}{2\mu g}$
The second box travels distance $s_2=\dfrac{(2u)^2}{2\mu g}=\dfrac{4u^2}{2\mu g}=4s_1.$
The two boxes together travel distance $s_1+s_2=5s_1=\dfrac{5u^2}{2\mu g}$
So what must be true for a collision to take place?

Sometimes multiple choice problems are not very well constructed.

 

[haruspex]

Hi @Jim Alexandridis and welcome to PF.

The first box travels distance $s_1=\dfrac{u^2}{2\mu g}$
The second box travels distance $s_2=\dfrac{(2u)^2}{2\mu g}=\dfrac{4u^2}{2\mu g}=4s_1.$
The two boxes together travel distance $s_1+s_2=5s_1=\dfrac{5u^2}{2\mu g}$
So what must be true for a collision to take place?

Sometimes multiple choice problems are not very well constructed.

Isn’t that the same as the OP did in post #1?

 

[kuruman]

Isn’t that the same as the OP did in post #1?

Yes. It's reassurance to a new user that when you're right, you're right.

 

[Jim Alexandridis]

I believe that for the collision to happen it has to be :

$S_{1}+S_{2}\ge S$

 

[kuruman]

I believe that for the collision to happen it has to be :

$S_{1}+S_{2}\ge S$

That is correct. As you already posted, this condition leads to $$u \geq \sqrt{\frac{2\mu~g~s}{5}}$$which is not one of the options (a) - (d). What you can do with this information is up to you. It ranges from doing nothing to bringing it to the attention of someone who is in a position to improve the quality control of exams like this.

 

[Hill]

That is correct. As you already posted, this condition leads to $$u \geq \sqrt{\frac{2\mu~g~s}{5}}$$which is not one of the options (a) - (d). What you can do with this information is up to you. It ranges from doing nothing to bringing it to the attention of someone who is in a position to improve the quality control of exams like this.

It might be a "copy/paste" issue. For example, in some other forum, I've just posted in LaTeX:

but after "edit and save" it has been converted to this: