How Long Does a Ball Take to Climb a Parabolic Ramp?

[errwrsysalan]

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!

 

[Simon Bridge]

What did you need the trig for?
Did you start with $\frac{1}{2}m(\dot x^2+\dot y^2) + mgy = \frac{1}{2}mu^2 : u=5,\; y=\frac{1}{3}x^2$ ?

 

[errwrsysalan]

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 learned to use and x and y things because I'm a high school 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!

 

[Simon Bridge]

Is this a problem set for high school?

 

[errwrsysalan]

Well. It's not, I made this question up. It'd be nice if you can solve it using high school knowledge only though.

 

[errwrsysalan]

Is this a problem set for high school?

I'm stuck again...

 

[errwrsysalan]

Is this a problem set for high school?

I might got it.. trying now

 

[errwrsysalan]

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?

 

[Simon Bridge]

Please answer the question?

 

[errwrsysalan]

Please answer the question?

Well. It's not, I made this question up. It'd be nice if you can solve it using high school knowledge only though.

I answered above. Basically this is some extracurricular work.

 

[Simon Bridge]

Sorry - misread.

 

[errwrsysalan]

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?

 

[Simon Bridge]

... 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.

 

[errwrsysalan]

... 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.