CLASSICAL MECHANICS & THERMODYNAMICS - Chapter 10 (Capstone), Exercise 2 Solution
==========================================================
Checking Safety for a Larger Loop

PROBLEM
-------
Suppose the loop's radius were increased to 4 m instead of 2.5 m,
with the car still entering the loop with 30,000 J of kinetic energy
at ground level (mass unchanged, 400 kg). Using conservation of
energy and the minimum-speed condition, determine whether the car
can safely complete this larger loop.

SOLUTION
--------
Step 1: Find the potential energy at the top of the new, larger loop
(height = 2r = 8 m).

    PE_top = mg(2r)
    PE_top = 400 x 9.81 x 8
    PE_top = 31,392 J

Step 2: Find the kinetic energy remaining at the top.

    KE_top = KE_launch - PE_top
    KE_top = 30,000 - 31,392
    KE_top = -1,392 J

A NEGATIVE kinetic energy is physically impossible - kinetic energy
can never be less than zero. This means the car does not have enough
total energy to even REACH the top of a 4 m radius loop while moving,
let alone maintain the minimum speed needed to stay on the track.

ANSWER: No, the car CANNOT safely complete a 4 m radius loop with
only 30,000 J of launch kinetic energy - it does not have enough
total energy to reach the top of the loop at all. The loop would
need to be smaller (as in the capstone's actual 2.5 m design), or the
car would need a larger initial kinetic energy, for the ride to work.

----
WHY THIS WORKS AS AN ANSWER

This exercise deliberately reverses the capstone's own successful
safety check to show what a FAILED design looks like. In the
capstone's real 2.5 m loop, converting to a height of 2r = 5 m left
10,380 J of kinetic energy at the top - enough to comfortably clear
the 4.95 m/s minimum speed requirement. Here, the taller loop (2r = 8
m) demands more potential energy (31,392 J) than the car's entire
30,000 J launch energy provides, so the car runs out of energy partway
up the loop and would fall away from the track well before reaching
the top - a genuine engineering failure that a real roller coaster
designer would have to catch during exactly this kind of calculation,
before construction, not after.