Answer:
W = 290.7 dynes*cm
Explanation:
d = 1/5 cm = 0.2 cm
The force is in function of the depth x:
F(x) = 1000 * (1 + 2*x)^2
We can expand that as:
F(x) = 1000 * (1 + 4*x + 4x^2)
F(x) = 1000 + 4000*x + 4000*x^2
Work is defined as
W = F * d
Since we have non constant force we integrate
W = [1000*x + 2000*x^2 + 1333*X^3] evaluated between 0 and 0.2
W = 1000*0.2 + 2000*0.2^2 + 1333*0.2^3 - 1000*0 - 2000*0^2 - 1333*0^3
W = 200 + 80 + 10.7 = 290.7 dynes*cm
1) The variable that is different in the two cases is , the duration of the collision
2) The change in momentum is the same in the two cases
3) The impulse is the same in the two cases
4) Case B will experience a greater force
Explanation:
1)
The variable that is different in the two cases is , the duration of the collision.
In fact, in the first case the wall is padded: this means that the collision will be "softer" and therefore will last longer, so the duration of the collision, , will be larger.
In the second case instead, the wall is unpadded: this means that the collision is "harder" and so it will last less time, therefore the duration of the collision will be smaller.
2)
The change in momentum in the two cases is the same.
In fact, the change in momentum is given by:
where:
m is the mass of the student
u is the initial velocity
v is the final velocity
In both cases, we have:
m = 75 kg
u = 8 m/s
v = 0 (they both comes to rest)
Therefore, the change in momentum is
3)
The impulse in the two cases is the same.
In fact, impulse is defined as the product of force applied, F, and duration of the collision, :
However, the force can be rewritten as product of mass (m) and acceleration (a), according to Newton's second law:
So the impulse is
The acceleration can be rewritten as rate of change of velocity:
So the impulse becomes
So, the impulse is equal to the change in momentum: and since in the two cases the change in momentum is the same, the impulse is the same as well.
4)
The force in the collision is related to the impulse by
where
J is the impulse
F is the force
is the duration of the collision
The equation can be rewritten as
In the two situations described in the problem (A and B), we already said that the impulse is the same (because the change in momentum is the same). However, in case A (padded wall) the time is longer, while in case B (unpadded wall) the time
is shorter: since the force F is inversely proportional to the duration of the collision, this means that in case B the student will experience a greatest force compared to case A.
Learn more about impulse:
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