Differential Equation Application Help?

[匿名]

2016-02-27 6:40 pm

Consider shooting a bullet of mass 𝑚 to three different big blocks. The resistance of the first block is assumed to be proportional to the bullet speed, that for the second block is proportional to 𝑣^(3/2), and that for the third block is proportional to 𝑣^2. If you fire at these blocks one by one with the same initial speed of 𝑣0 for your bullets, compute the farthest the bullet can travel in each of these all three cases. Note that the gravitational force for this problem is negligible.

回答 (1)

az_lender

2016-02-27 7:33 pm

✔ 最佳答案

What is "resistance" ? If it is a force, the deceleration of the bullet is directly proportional to it. So you have:
m dv/dt = -kv (block 1),
m dv/dt = -kv^(3/2)...(block 2),
and m dv/dt = -kv^2 (block 3).

Block 1 solution:
m dv/v = -k dt =>
v = v0 * e^(-kt/m).
The distance traveled is the integral from t = 0 to infinity of
v0*e^(-kt/m) = v0*(m/k).

Block 2 solution:
m dv/[v^(3/2)] = -k dt =>
(-2m)/v^(1/2) = -kt + C =>
v^(1/2) = 2m/(kt - C) = >
v = 4m^2/(kt - C)^2 where C = -4m^2/v0.
Distance = integral from t = 0 to infinity of v dt.
I leave it to you to complete this case.

Block 3 solution:
m dv/v^2 = - k dt =>
- m/v = -kt + C =>
C = -m/v0 =>
v = m/(m/v0 - kt).
Again the distance is the integral from t = 0 to infinity of v dt,
and again I leave it to you to complete that.

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