A particle falls vertically from rest. At speed v v\,v m s−1^{-1}−1 it experiences a resistive force of magnitude mv225\dfrac{mv^2}{25}25mv2 N, where m m\,m kg is its mass. Take downwards as positive.
Show that dvdt=g−v225\dfrac{dv}{dt}=g-\dfrac{v^2}{25}dtdv=g−25v2.
By separating variables, show that 52gln(5g+v5g−v)=t\displaystyle \frac{5}{2\sqrt g}\ln\left(\frac{5\sqrt g+v}{5\sqrt g-v}\right)=t2g5ln(5g−v5g+v)=t.
Hence express v v\,v explicitly in terms of t t\,t and ggg.
State the limiting speed.
92 exam-style questions on WJEC A Level Maths 4.6 Integration (A-level only), covering 4.6.1 Integration (A-level only) and 4.6.2 Integration (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.