Integral Solve for u + at^2/c^2

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In summary, the conversation discusses solving a complicated integral by making a substitution. After making the substitution and correcting the limits of integration, the integral simplifies to a more manageable form that can be easily solved.
  • #1
jcsd
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Can someone solve this integral as the answer I get looks suspicously complicated:

[tex]\int^{0}_{\frac{-u}{a}} t\sqrt{1 - \frac{(u + at)^2}{c^2}} dt [/tex]
 
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  • #2
Make the substitution [itex]\frac{u + at}{c} = \cos \theta[/itex]. You should be able to get it down to this:

[tex]\frac{c}{a^2} \left (u\int _{\frac{\pi}{2}} ^{\arccos \left(\frac{u}{c}\right)} \sin ^2 \theta d\theta - c\int _{\frac{\pi}{2}} ^{\arccos \left(\frac{u}{c}\right)} \sin ^2 \theta \cos \theta d\theta \right )[/tex]

And you can easily solve that on your own.

EDITED to fix limits of integration as per HallsOfIvy's comment.
 
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  • #3
The substitution might work but the limits of integration are wrong. When t= 0, cos[theta]= u/c so [theta]= cos<sup>-1</sup>(u/c). When t= u/c, cos[theta]= 0 so [theta]= [pi]/2.
 
  • #4
Thanks for that, I realized I made a slight error so it became slightly easier to solve.
 

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"Integral Solve for u + at^2/c^2" is an equation that is used to find the value of a variable, u, in terms of time (t) and the acceleration (a) and speed of light (c) constants. This equation is commonly used in physics and engineering for solving problems involving motion and acceleration.

2. How is the equation "Integral Solve for u + at^2/c^2" derived?

The equation "Integral Solve for u + at^2/c^2" is derived from the integral of the equation for velocity, which is v = u + at. By taking the integral of this equation with respect to time, we can solve for the variable u in terms of t, a, and c. This derivation is based on the fundamental principles of calculus.

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