This is an unusual question about an elastic collision so i have two carts and one of them
has a spring on it and that's going to make it so the carts bounce off of each other and they
each will have a their own final velocity and i solved that sort of ordinary problem in another
video already and i'll just post a link to that but in this problem we want to know what's the
maximum compression of that spring while the collision is actually occurring and this is for
the most part just a concept question because once you figure it out there's hardly any math to do
so we have to imagine what's happening when the collision begins the two kilogram cart will be
moving faster than the 1.5 and squishing the spring as a result and then at some point the
1.5 kilogram mass starts to bounce off which means it's now moving faster than the 2 kilogram mass so
what's right in between those two the 2 kilogram approaching the 1.5 and then the 1.5 running away
from the 2 kilogram there's a special moment where they're actually moving at exactly the same speed
and that allows me to very quickly solve this using momentum conservation
so at this special moment they're sharing the same speed uh the distance between them is switching
between getting smaller and getting bigger so right at this moment the distance between them is
not changing at all and they have the same speed and when i apply momentum conservation to this
it's as simple as an inelastic collision i would just say p initial equals p final
and my initial momentum was 2 kilograms times three meters per second and my final momentum
is three and a half kilograms all moving at the same speed
a v final and i've got v final immediately and i get 1.715 meters per second
well how is this going to get us where we need to go the key here
is that i know that the total energy in the initial state is going to be equal to the
energy in the final state because it's a perfectly elastic collision i'm using a spring to do that
so what i'm going to do is find out if i just look at how fast these masses are going in the
final state how much energy is missing and then i'm going to say that the missing energy is at
this moment stored in the spring and then i should be able to find the compression so
my initial kinetic kinetic energy for the system one half times the mass times the speed squared
and i get nine joules out of that
my final kinetic energy is one half times how much mass is moving that's three and a half kilograms
and multiplied by the final speed
1.714 meters per second squared and out of that i get 5.14 joules of energy
which means 3.86 joules of energy
is not in the motion of the masses but is instead stored in the compression of the spring
now i go to my spring energy formula potential energy in a spring is one half
kx squared i know the potential energy in there is 3.86 joules
i've got a spring constant of 500 newtons per meter all that's left is that compression that
i'm after and i do a little bit of work in my calculator and i get .124 meters or
12.4 centimeters of compression
so i've always liked this problem because it allows us to get into some of the details of
what's actually going on during one of these springy collisions