Particle Motion

**xxlvh** · December 28th 2008, 07:11 PM

A particle moves along the x-axis so that at any time t > 0, its acceleration is given by a(t) = ln(1 + 2^t). If the velocity of the particle is 2 at time t = 1, what is the velocity of the particle at time t - 2?
- I'm having a tough time getting the antiderivative for a(t)

**mr fantastic** · December 28th 2008, 09:30 PM

Originally Posted by xxlvh

A particle moves along the x-axis so that at any time t > 0, its acceleration is given by a(t) = ln(1 + 2^t). If the velocity of the particle is 2 at time t = 1, what is the velocity of the particle at time t - 2?
- I'm having a tough time getting the antiderivative for a(t)

Do you mean $a(t) = \ln (1 + 2^t)$ ?

**xxlvh** · December 29th 2008, 07:28 PM

Whoops! Yes that is what I meant to put.

**mr fantastic** · December 30th 2008, 12:55 AM

Originally Posted by mr fantastic

Do you mean $a(t) = \ln (1 + 2^t)$ ?

Originally Posted by xxlvh

Whoops! Yes that is what I meant to put.

Is an exact answer required? Because you can express the velocity as

$v(t) = \int_1^t \ln (1 + 2^u) \, du + 2$ .

Then

$v(2) = \int_1^2 \ln (1 + 2^u) \, du + 2$

and the integral can be approximately evaluated using technology.

I don't think an exact answer in terms of a finite number of elementary functions is possible (although I may stand corrected on this).

**Mathstud28** · December 30th 2008, 01:28 AM

Originally Posted by mr fantastic

Is an exact answer required? Because you can express the velocity as

$v(t) = \int_1^t \ln (1 + 2^u) \, du + 2$ .

Then

$v(2) = \int_1^2 \ln (1 + 2^u) \, du + 2$

and the integral can be approximately evaluated using technology.

I don't think an exact answer in terms of a finite number of elementary functions is possible (although I may stand corrected on this).

I agree with Mr. F. The symbolic answer to this problem involves the dilogarithim function...which has absolutely no meaning or use here, but thanks to Mathcad we can approximate the answer to $\int_1^2 \ln\left(1+2^x\right)~dx\approx 1.346$ . Hope that helps.

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