Can bullets and planks madden you? Maybe yes!

A bullet starts with some initial velocity uu.

It hits a plank which reduces it's velocity by ux\dfrac{u}{x}.

Let nn be the number of planks required to stop it.

So on further calculations, I knew that n=x22x1 n = \dfrac{x^2}{2x-1}

Is it correct? I am unsure as

If x=20x = 20

Then n=11n = 11 approx.

If x=50x=50

Then n=26n = 26 approx .

This means that the larger the velocity the plank will decrease, the larger will be the no. of planks required. This is quite weird as it should be the opposite.

Now, this is how I got my formula.

Note: All symbols have their usual meaning like v,u,av , u , a and ss.

Let the no. of planks required be nn

Let a plank decrease it's velocity uu by ux\dfrac{u}{x}

Let thickness of one plank be zz

After crossing the first plank , v=uuxv = u - \dfrac{u}{x}

Now, by third equation of motion for the first plank,

v2=u2+2asv^2 = u^2 + 2as

(uux)2=u2+2az\left(u - \dfrac{u}{x}\right)^2 = u^2 + 2az

(uux)2u2=2az\left(u - \dfrac{u}{x}\right)^2 - u^2 = 2az

(uxux)2u2=2az\left(\dfrac{ux - u}{x}\right)^2 -u^2 = 2az

(ux)2+u22u2x(ux)2x2=2az\dfrac{(ux)^2 + u^2 - 2u^2 x - (ux)^2}{x^2} = 2az

u22u2x2x2z=a\dfrac{u^2 - 2u^2 x}{2x^2 z} = a

Now after crossing nn planks, it's velocity will be 0.

Now by again applying third equation of motion we get,

v2=u2+2asv^2 = u^2 + 2as

Here, v=0v = 0

0=u2+2(u22u2x2x2z)(n)(z)0 = u^2 + 2 (\dfrac{u^2 - 2u^2 x}{2x^2 z}) (n)(z)

0=u2+u22u2xx2(n)0 = u^2 + \dfrac{u^2 - 2u^2 x}{x^2} (n)

0=(ux)2+(u22u2x)nx20 = \dfrac{(ux)^2 + (u^2 - 2u^2 x)n}{x^2}

0=u2x2+(u22u2x)n0 = u^2 x^2 + (u^2 - 2u^2 x)n

0=u2(x2+[12x]n)0 = u^2(x^2 + [1 - 2x]n)

0=x2+(12x)n0 = x^2 + (1 - 2x)n

x2=(12x)n-x^2 = (1 - 2x)n

x212x=n\large\dfrac{-x^2}{1 - 2x} = n

      OR

n=x22x1\large n=\dfrac{x^2}{2x - 1}

I hope that this is correct.

Note by Vinayak Bansal
1 year, 8 months ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

There are no comments in this discussion.

×

Problem Loading...

Note Loading...

Set Loading...