in this video we're analyzing the impact of a dart with a block then the dart and block
combination compresses a spring to its maximum compression distance and this setup can actually
be used to measure the initial velocity of the dart based on the maximum compression of the
spring and that's the point of the problem we're going to work through the Collision backwards
from the maximum compression of the spring all the way back to the initial speed of the dart
so in part A we're asked to compute the speed of the combined masses just after the impact in
other words we're comparing the second picture to the third picture and working our way backwards
we have the maximum compression of that spring storing spring potential energy that came from
the kinetic energy of the combined masses after the impact being converted into spring potential
energy and we'll go ahead and call that velocity V so energy is conserved in this part of the process
and in my initial State I have the kinetic energy of the combined masses I'll call that one-half
Little M for the dart Big M for the block so M plus M times V squared that's all converted
into the spring potential energy at the maximum compression so that's one half k x squared the one
halves cancel here and I can divide by M plus M on both sides and square root the result to
solve for the velocity of those combined masses as the compression process begins so I find that
V is the square root of KX squared divided by the sum of the two masses Little M plus Big M
so it's time to put in the numbers and we were told the spring constant was 10 Newtons per meter
the maximum compression was 3.8 centimeters which needs to be converted to meters so that's 0.038
remember to square that and then the sum of the two masses well that's 30 grams for the dart and
100 grams for the block so 130 grams or 0.130 kilograms when we run the numbers on this to
three significant digits we get 0.333 meters per second for the velocity of the combined
masses after the impact has occurred next we want to compute the speed of the dart before
the impact occurred so the key physical Point here is to realize that we had an inelastic collision
between the dart and the block and provided that we don't have too much compression of the spring
happening yet during the Collision itself the momentum is going to be conserved during this
collision and that's a really good approximation the Collision should happen very rapidly before
the spring starts to compress and exert a force on the combined masses so if I call the first picture
state number one for the system I could express this as P1 is equal to P2 the momentum in state 1
is equal to the momentum in state two keep in mind that energy is lost in an inelastic collision so
a common mistake you could make in this problem is comparing the energy in state two or three to
the energy in state one energy is not conserved in that inelastic collision so it's not going to
work out so our physical handle on things is that the momentum is conserved and I can say
in state one the Little M the mass of the dart times V naught the initial velocity of the dart
that's all the momentum in the system in that state that's going to be equal to the momentum
in the final state where the masses are combined into Little M plus Big M with a velocity of V that
we computed in part A so now we can just solve for V naught the initial velocity of the dart
and that's Little M plus Big M the combined masses times the speed after the Collision over Little M
the mass of the dart and it's time to plug in the numbers remember the combined masses was
a total of 130 grams or 0.130 kilograms then the mass of the dart that's .030 kilograms and
the speed of the combined masses after the Collision we computed that in part A 0.333
meters per second when we run the numbers on this to three significant digits we get 1.44 meters
per second for the initial velocity of the dart finally we're asked to compute the energy lost
in the collision and the reason I put this on this test question was to remind everybody that energy
is lost in the Collision so it's trying to help the students avoid the mistake of comparing the
initial energy to the final energy you're going to see a loss of energy in that inelastic collision
so all we have to do to answer the question is compute the initial kinetic energy that's one half
little m v naught squared so the mass of the dart 0.030 initial velocity of the dart squared so 1.44
meters per second all squared and when we run the numbers we get .0311 joules then we compute the
final energy and you have a choice here we could compute the final energy as the kinetic energy of
the combined masses before the spring compresses or we could use the energy stored in the spring
at maximum compression remember the spring compression process is energy conserving so either
one of those is going to give the same answer I'm going to go ahead and use the kinetic energy
immediately after the Collision so our combined masses were 0.130 kilograms the speed just after
the impact 0.333 meters per second and when we run the numbers on this we get .0072 joules so
how much was lost you just take the difference between the initial kinetic energy and the final
kinetic energy and this turns out to be .0239 joules and we're done if you enjoyed this video
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