Evaluate t = (1/3.6) times the integral from 0 to 7.07 of (1.1 x 9.13) / (10.752 − 9.13 x 9.8 x 0.01 − (0.04985 x 0.2032 x v²)/21.15) dv.

Step-by-step solution

1. 1

   Collapse every constant before doing any calculus. The integrand looks formidable only because the constants are unmultiplied. Numerator:

   $1.1\times 9.13=10.043$

   Constant part of the denominator:

   $10.752-9.13\times 9.8\times 0.01=10.752-0.89474=9.85726$

   Coefficient of $v^{2}$:

   $\frac{0.04985\times 0.2032}{21.15}=\frac{0.01013052}{21.15}=4.78937\times 10^{-4}$

2. 2

   Rewrite in a recognisable standard form.

   $t=\frac{10.043}{3.6}\int_{0}^{7.07}\frac{dv}{9.85726-4.78937\times 10^{-4}v^{2}}$

   Factor the $v^{2}$ coefficient out of the denominator to expose the difference-of-squares shape:

   $=\frac{10.043}{3.6\left(4.78937\times 10^{-4}\right)}\int_{0}^{7.07}\frac{dv}{a^{2}-v^{2}},\qquad a=\sqrt{\frac{9.85726}{4.78937\times 10^{-4}}}=143.463$

3. 3

   Use the logarithmic antiderivative. Partial fractions give

   $\int\frac{dv}{a^{2}-v^{2}}=\frac{1}{2a}\ln\left|\frac{a+v}{a-v}\right|+C$

   (equivalently $\tfrac1a\operatorname{artanh}\tfrac va$). The upper limit $7.07$ is far below $a=143.463$, so $a^{2}-v^{2}>0$ throughout and the integrand never blows up.

4. 4

   Evaluate the definite integral.

   $\int_{0}^{7.07}\frac{dv}{a^{2}-v^{2}}=\frac{1}{2(143.463)}\ln\frac{143.463+7.07}{143.463-7.07}=\frac{\ln(1.10367)}{286.926}=3.4379\times 10^{-4}$

5. 5

   Multiply by the constant in front.

   $\frac{10.043}{4.78937\times 10^{-4}}\times 3.4379\times 10^{-4}=20969\times 3.4379\times 10^{-4}=7.2091$

   so the integral itself is $7.2091$, and

   $t=\frac{7.2091}{3.6}=2.0025$

   $\boxed{t\approx 2.00\ \text{s}}$

6. 6

   Sanity-check with the constant-integrand estimate. Because $v$ only ever reaches $7.07$ while $a=143.5$, the $v^{2}$ term shifts the denominator by at most $4.79\times 10^{-4}(7.07)^{2}=0.0239$ out of $9.857$ — about $0.24\%$. Treating the integrand as the constant $\tfrac{10.043}{9.85726}=1.01885$ gives $t\approx\tfrac{1.01885\times 7.07}{3.6}=2.0009$ s, within $0.1\%$ of the exact $2.0025$ s ✓.