1. Aug 14, 2005

### altecsonyc

Can Someone please help me with this problem! I need some hints how to solve this problem, i am already stuck on part A :(.
This is my practice assignment for the final coming up next 2 weeks. Unfortunately, the professor does not provide solution.

THE PICTURE OF THIS PROBLEM IS ILLUSTRATED BY THE LINK BELOW.

Starting from the initial position (the origin of the coordinate system) a cart with a mass of 2000 kg is accelerated from the initial velocity zero by acceleration a1 which is uniform over a segment of 40 m. Following this segment, the cart proceeds through a turn (20 m radius), is then accelerated again over a segment of 10 m length, coasts toward the top of a hill where its velocity is near zero. The cart then rolls down to ground level, through a circular valley (radius 10 m) and along a straight segment of 2 m length, before leaving the track in free fight, initially inclined at 45 degree. In order to ensure a safe landing, an angular impulse is exerted on the cart just before leaving the track, causing the cart to rotate by a constant angular velocity w, during the flight. This rotation is necessary to have the cart axis aligned with the track at the landing point. Departure and landing occur at the same vertical position. On a short segment of 5 m the cart is decelerated by a3, before coasting through the second turn (20 m radius) and it is further decelerated uniformly (a4) over the final 30 m to come to a full stop at the origin of the coordinate system.

Determine the following design parameters:
a) Appropriate accelerations a1, a2, to meet the following requirements:
_ The cart must come to a stop on top of the hill.
_ The magnitude of the acceleration through the first turn must not be larger than 3g.
b) The coordinates of the point where the cart departs into free flight.
c) The velocity of the cart at the point of departure.
d) The coordinates of the landing point and the flight path angle (_2) at that point.
e) The duration of the flight, and the angular velocity w which is required to have the cart aligned with the track at landing.
f) The velocity of the cart as it reaches ground level (just before the 5m deceleration segment).
g) Appropriate accelerations a3, a4, to meet the following requirements:
_ The cart must come to a full stop at the origin of the coordinate system.
_ The magnitude of the acceleration through the second turn must not be larger than 3g.
h) The duration of the entire ride.

Last edited by a moderator: Aug 14, 2005
2. Aug 14, 2005

### Clausius2

As good engineers, traditionally here nobody gets his hands dirty doing numbers or solving homeworks. Unless you find a brave man here, you will do it better by posting your problem in homework section. Good luck.

3. Aug 14, 2005

### brewnog

You'll get much better reception here if you post what work you've done, and show us where you've got stuck. Good luck!

4. Aug 14, 2005

### Staff: Mentor

Well all the solutions come from basic kinematics.

"The cart must come to a stop on top of the hill." where the gravitational potential energy is maximum. Conservation of energy.

I don't see any mention of friction or wind resistance (which keeps it simple). Instead there is mention of acceleration and deceleration.

Also for angular acceleration $\alpha$ = $\omega^2 r$.

As for the jump, think equations for trajectory (parabolic). Some mass departs a point at some angle $\theta$ with respect to the horizontal and initial velocity, and then lands at some other point (at what angle?).

And this does belong in the college homework section.

5. Aug 14, 2005

### altecsonyc

For part A.
Knowing V0=0, X-X0=40=X1, X2=10
I have (V1)^2=(V0)^2 + 2*(a1)(X-X0)
(V1)^2= 2*(a1)*(X1)

and on top of the loop V = 0

SO m*g*h = 1/2*m*(V3)^2
(V3)^2 = 2*g*h
(V3)^2 = V2^2 + 2(a2)*(X2)

V1 = V2 (since, it's circular uniform)
Ac = V^2/R

I am stuck here, i dont know what to do next to find the a2 and a1, and V2. I think, im missing something, i have 2 equations with 3 unknows..but i tried for many hours and still couldn't figure it out.

So do you guys think my concept is correct? for some reason the formular Ac= V^2/R still haven't use yet, which i think there must be something wrong with my equation. I believe Ac=V^2/R must something to do with the V2 and somehow relate to the a1 and a2.
Thank you

6. Aug 14, 2005

### altecsonyc

Oh, i have thought about the 3g think.
so the acceleration through the first turn must not be larger than 3g.
SO

A = 3g =< sqrt(At ^2 + Ac^2)

but At = (V1)^2 / (2*X1)
Ac = (V1)^2/ R

solve for V1, and V1=V2

Does this make sense?