What is the longest time that a particle can take in travelling between two points if it never increases...

analysis/particle

Problem

What is the longest time that a particle can take in travelling between two
points if it never increases its acceleration along the way and reaches the
second point with speed V?

Show solution

Assumptions:

1. x(0) = 0; x(T) = X

2. v(0) = 0; v(T) = V

3. d(a)/dt <= 0

Solution:

a(t) = constant = A = V^2/2X which implies T = 2X/V.

Proof:

Consider assumptions as they apply to f(t) = A * t - v(t):

1. integral from 0 to T of f = 0

2. f(0) = f(T) = 0

3. d^2(f)/dt^2 <= 0

From the mean value theorem, f(t) = 0.  QED.