# Constraints on Acceleration Given Endpoints of Motion

1. May 11, 2013

### Kakashi24142

1. The problem statement, all variables and given/known data
A particle of mass M moves on the X-axis as follows: It starts from rest at t = 0 from the point x = 0 and comes to rest at t = 1 at the point x = 1. No other information is available about it smotion at intermediate times (0 < t < 1). If α denotes the instantaneous acceleration of the particle, then prove that |α| must be ≥ 4 at some point in its path.

2. Relevant equations
$$\int_0^1{v(t)\,dt} = x(1) - x(0) = 1\\ \int_0^1{a(t)\,dt} = v(1) - v(0) = 0\\$$

3. The attempt at a solution
If constant accelerations of equal magnitudes are assumed for the period of acceleration and the period of deceleration, one obtains that the acceleration is 4 for the first 1/2 second and -4 for the last 1/2 second. I can see why the statement is true given these calculations, but could someone suggest a more rigorous way to prove this?

2. May 11, 2013

### Staff: Mentor

If |α| <=4, what is the quickest way to reach x=1? You can use symmetry to consider the acceleration part only, if you like.
If |α| <4, can you still have the same time?