Acceleration Proportional to
Displacement

## Question:

The acceleration of a particle is defined by the relation
a=6*x-14 feet per second per second. Knowing that v=4 feet per
second at x=0, determine the maximum value of x and the velocity
when the particle has traveled a total distance of 1 foot.
## Answer:

One interpretation of the question is that acceleration is zero
at x=2.33 feet and varies in proportion to x (6*x) away from that
point being negative when x is less that 2.33 and positive when x
is greater than 2.33. When acceleration varies linearly with
position, like a spring was pushing against the moving particle.
The initial condition we have is velocity 4 at position zero so
the particle, being left of 2.33 is subject to an acceleration in
the negative direction, slowing it down. We know that eventually
it gets stopped and turned around by the acceleration, otherwise
x would not have a maximum, but keep increasing forever. The
basic question is, "How far will the particle have to go to
suffer a 4f/s change in velocity?"
The only approach that occurs to me at the moment is to try
something with the conservation of energy. The force exerted by
our hypothetical spring is the acceleration of the particle
divided by its mass. We do not know the mass but let's press
on a bit. The work, W, done in moving the particle from x to x+dx
is the distance dx times the force it must overcome. This work
must come from the kinetic energy, ke, of the particle which is
1/2*m*v^{2}. When all the kinetic energy is gone, the
particle will stop. Initial ke is
1/2*m*4^{2} at x=0.

The work done as a function of x is the sum of all the little
dw from x=0 to x. The work dw is the force at x times dx.
dw=6*m*(x-2.33)*dx so by integrating over x,
W=6*m(1/2x^{2}-2.33*x). When W=ke we will be at
the maximum value of x.
6*m*(1/2*x^{2}-2.33x)=1/2*m*16. Solve for x. The
m's cancel out so 3*x^{2}-7*x=8 or
3*x^{2}-7x-8=0. I get x=0.833
and x=3.16 as possible solutions. Rejecting the
solution that is on the wrong side of 2.33, we get
x=.833 for the answer to part a.

For part b, the velocity is zero at .833 ft so we need to know
how much velocity is gained in the negative direction in the
first 0.167 feet. The work done by the force will again equal the
gain in kinetic energy. I leave it to you to do the
arithmetic.

This information is brought to you by M. Casco Associates, a
company dedicated to helping humankind reach the stars through
understanding how the universe works. My name is James D. Jones.
If I can be of more help, please let me know. JDJ