free falling

2015-05-09 9:52 pm

Note tha v = v∞(1-e^-γt)

can you explain why
1. γt >> 1 ( what is the meaning of >> instead of >?)
→e^-γt～0?

2. γt << 1
→e^-γt～1-γt??

更新1:

Im sorry the topic shouldnt be free falling, it should be changing acceleration

更新2:

sorry,still have a little problem. An object falls under gravity and air resistance force bv. It is known there is a characteristic time τ=m/b such that the object comes close to the terminal velocity for t>>τ. Derive an exact expression for the distance d travelled at time t. Ans: d = mgτ/eb

更新3:

Ive just differentiated v=v∞(1-e^-γt) and get -mg/be ,why wrong? ~_~

回答 (1)

天同

2015-05-09 10:13 pm

✔ 最佳答案

1. The symbol ">>" means "much much greater than".

Because e^(-γt) = 1/e^(γt)
when (γt) is large, 1/e^(γt) will be small. Hence, when e^(-γt) approaches zero when (γt) is very large.

2. Using the exponential series,
e^(-γt) = 1 - γt + (γt)^2/2! - (γt)^3/3! + .....
When γt << 1, the higher order terms can be neglected, hence,
e^(-γt) = 1 - γt

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