Direct collisions on an inclined plane

  • Thread starter Woolyabyss
  • Start date
  • #1
143
1

Homework Statement



A smooth sphere falls vertically and strikes a fixed smooth plane inclined at an angle of ∅ to the horizontal.If the coefficient of restitution is (2/3) and the sphere rebounds horizontally,
Its speed before impact is u and after impact v
calculate the fraction of kinetic energy lost during impact.


Homework Equations





The Attempt at a Solution



Diagram of the question is attached to this post.

i = horizontal component vector and j = vertical component vector

Taking the inclined plane as the x axis
v, vcos∅i + vsin∅j ..... alternate angle to inclined angle

u, vcos(90 - ∅ )i - v sin(90 - ∅)j = usin∅i - ucos∅j


the I component remains the same

vcos∅ = usin∅

v = utan∅

vsin∅/(ucos∅) = 2/3 (v/u)tan∅ = 2/3 (utan∅/u)tan∅ =2/3

(tan∅)^2 = 2/3 tan∅ = (√6)/3


v = u(√6)/3

.5mu^2 - .5mv^2 = energy loss

.5mu^2 -.5m(6/9)u^2 = .5mu^2 - m(6/18)u^2 = .5mu^2 - (1/3)mu^2 = (1/6)mu^2

fraction of kinetic energy lost is 1/6


my book says the answer is 1/3 but I got 1/6. Any help would be appreciated.
 

Attachments

  • Y0nW0sFi72ZnOGbAKpEQYekz8QTIQuHdbGR3-84i25c.jpg
    Y0nW0sFi72ZnOGbAKpEQYekz8QTIQuHdbGR3-84i25c.jpg
    18.1 KB · Views: 362
Last edited:

Answers and Replies

  • #2
Andrew Mason
Science Advisor
Homework Helper
7,671
393
vcos∅ = usin∅

v = utan∅
Correct so far

vsin∅/(ucos∅) = 2/3 (v/u)tan∅ = 2/3 (utan∅/u)tan∅ =2/3
It is a bit convoluted there. I can't follow what you have done. Just use:

vsin∅/(ucos∅) = 2/3

so: tan∅ = 2/3(u/v)

Since you also have tan∅ = v/u

∴ v^2/u^2 = 2/3

AM
 
  • #3
143
1
I used the equation

v^2/u^2 = 2/3

v^2 = (2/3)u^2

so .5mu^2 - .5m(2/3)u^2 = 1/6mu^2

fraction of kinetic energy lost during impact = 1/6

I got the same answer do you think my book could be wrong?
 
  • #4
haruspex
Science Advisor
Homework Helper
Insights Author
Gold Member
2020 Award
36,422
6,952
so .5mu^2 - .5m(2/3)u^2 = 1/6mu^2
fraction of kinetic energy lost during impact = 1/6
I got the same answer do you think my book could be wrong?
What fraction is 1/6mu^2 of .5mu^2?
 
  • #5
143
1

Related Threads on Direct collisions on an inclined plane

Replies
7
Views
2K
  • Last Post
Replies
9
Views
2K
Replies
1
Views
560
  • Last Post
Replies
3
Views
630
  • Last Post
Replies
8
Views
1K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
2
Views
2K
Top