• Support PF! Buy your school textbooks, materials and every day products Here!

A ball rolling up a parabolic ramp

  • #1
1. Assume there is gravity and no external force acting on the system. A ball has an initial velocity of 5 m/s and climbs up a parabolic ramp, which is defined by y=(x^2)/3. If the ball rolls exactly along the path of ramp and energy of the ball is conserved, starting from (0,0), calculate the time taken for the ball to climb to a height of y=1.

2. Ek+Ep=Einit, Vy=||v||sin(arctan(dy/dx))

3. I made a differential equations using the formula above but the differential I got is only numerical integratable (results given by wolfram alpha). I'm looking for an easier way to solve this problem. So, guys, please help! Thanks!
 
Last edited:

Answers and Replies

  • #2
Simon Bridge
Science Advisor
Homework Helper
17,847
1,644
What did you need the trig for?
Did you start with ##\frac{1}{2}m(\dot x^2+\dot y^2) + mgy = \frac{1}{2}mu^2 : u=5,\; y=\frac{1}{3}x^2## ?
 
  • #3
What did you need the trig for?
Did you start with ##\frac{1}{2}m(\dot x^2+\dot y^2) + mgy = \frac{1}{2}mu^2 : u=5,\; y=\frac{1}{3}x^2## ?
That seems pretty helpful! I haven't learnt to use and x and y things because I'm a highschool student, but I've done stage 1 college maths so I think I can cope with that. I'll give it a try! Thank you!
 
  • #4
Simon Bridge
Science Advisor
Homework Helper
17,847
1,644
Is this a problem set for high school?
 
  • #5
Well. It's not, I made this question up. It'd be nice if you can solve it using highschool knowledge only though.
 
  • #6
  • #7
Is this a problem set for high school?
I might got it.. trying now
 
  • #8
Is this a problem set for high school?
I'm trying now, but still, can you show me how you would do it please?

Guess what. I used the formula you gave me and that ends up the same differential equation I got last time using trig, which I need to use a computer to calculate it. Can you help me?
 
  • #9
Simon Bridge
Science Advisor
Homework Helper
17,847
1,644
Please answer the question?
 
  • #10
Please answer the question?
Well. It's not, I made this question up. It'd be nice if you can solve it using highschool knowledge only though.
I answered above. Basically this is some extracurricular work.
 
  • #11
Simon Bridge
Science Advisor
Homework Helper
17,847
1,644
Sorry - misread.
 
  • #12
Sorry - misread.
It's ok. So, right now I used your formula, and it ended up exactly the same as last time I did it using trig.
I got dt/dy=0.5sqrt((3+4y)/(25y-2gy^2)) dy, which even wolfram alpha couldn't give an exact solution.
I believe there is a better way to do this. Can you show me please?
 
  • #13
Simon Bridge
Science Advisor
Homework Helper
17,847
1,644
... this is a specific case of an object sliding inside a parabolic bowl.
If you google for that, you'll get a bunch of examples. i.e. See attachment below.

Note: ##\dot x = v(x)## is where you get stuck.
You can do $$T = \int_0^{\sqrt 3} \frac{dx}{v}$$ ... i.e. what you were doing but as a definite integral.
The indefinite integral involves elliptical functions.

It is not unusual for quite simple-seeming setups to be very difficult to solve - or even have no analytic solutions at all.
ie. Try working out the exact equation of motion for a simple pendulum - or a puck sliding in a spherical bowl.
 

Attachments

  • #14
... this is a specific case of an object sliding inside a parabolic bowl.
If you google for that, you'll get a bunch of examples. i.e. See attachment below.

I'd usually reach for lagrangian mechanics.
It is not unusual for quite simple-seeming setups to be very difficult to solve - or even have no analytic solutions at all.
Try working out the exact equation of motion for a simple pendulum for example - or a puck sliding in a spherical bowl.
Thank you! No problem. I need to sleep as well. I'll have a look tomorrow.
 

Related Threads on A ball rolling up a parabolic ramp

Replies
4
Views
18K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
0
Views
1K
  • Last Post
Replies
4
Views
3K
  • Last Post
Replies
3
Views
3K
Replies
2
Views
5K
Replies
4
Views
3K
Replies
16
Views
2K
Replies
6
Views
11K
Replies
3
Views
3K
Top