Question

Starting from rest a particle moves in a straight line with acceleration
$a=(25-t^2{)}^{1/2}m/s^{2}for0\le t\le 5s$
$a=\frac{3\pi }{8}m/s^{2}fort\ge 5s$
The velocity of particle at t = 7s is:

• A. 11 m/s
• B. 22 m/s
• C. 33 m/s
• D. 44 m/s

Answer 1

The correct option is B 22 m/s

$v=\int adt$

$a=(25-t^2{)}^{1/2}m/s^{2}for0\le t\le 5s$
$=\frac{3\pi }{8}m/s^{2}fort\ge 5s$

$v=\int (25-t^2{)}^{1/2}dt+a0\le t\le 5s$
$=\int \frac{3\pi }{8}dt+bt\ge 5s$

$v=\frac{t}{2}\sqrt{5^2-t^2}+\frac{25}{2}si{n}^{-1}\left(\frac{t}{5}\right)+a0\le t\le 5s$
$=\frac{3\pi t}{8}+bt\ge 5s$

at $t=0,v=0$ gives $a=0$

finding $b$ using the fact that speed is a continous fuction

$li{m}_{t\to 5^{-}}v=li{m}_{t\to 5^{+}}v$

$0+\frac{25}{2}si{n}^{-1}\left(\frac{5}{5}\right)=\frac{3\pi \times 5}{8}+b$

$b=\frac{35\pi }{8}$

$v=\frac{t}{2}\sqrt{5^2-t^2}+\frac{25}{2}si{n}^{-1}\left(\frac{t}{5}\right)0\le t\le 5s$
$=\frac{3\pi t}{8}+\frac{35\pi }{8}t\ge 5s$

so at $t=7$

$v=\frac{3\pi \times 7}{8}+\frac{35\pi }{8}$

$v=\frac{8\pi \times 7}{8}$

$v=7\pi$

$v=22m/s$