A particle of mass m is able to move along the x axis and is subject to a force
F = −ax2,
where a is a positive constant The particle is projected from x = 0 with a
velocity x dot = (v0) > 0.
Use the energy equation to show that the particle comes to rest at
x = (3m(v0)^2/2a)^1/3.
Energy
2009-11-19 4:33 am
回答 (1)
2009-11-19 5:41 am
✔ 最佳答案
Method I: Energy equation:Work done by the force = K.E. lost by the particle
S F dx = 1/2 mv0^2
S -ax^2 dx (from 0 to x) = 1/2 mv0^2
-a[x^3/3] (from 0 to x) = 1/2 mv0^2
ax^3 / 3 = 1/2 mv0^2
x = [3mv0^2 / 2a]^1/3
Method II: Force equation:
F = -ax^2
mx" = -ax^2
mv dv/dx = -ax^2
Separating variables,
m S v dv (from v0 to 0) = -a S x^2 dx (from 0 to x)
m[v^2/2] (from v0 to 0) = -a[x^3/3] (from 0 to x)
mv0^2 / 2 = ax^3 / 3
The required distance, x = [3mv0^2 / 2a]^1/3
參考: Physics king
收錄日期: 2021-04-19 20:40:50
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