Complete question:
The acceleration of a particle traveling along a straight line is a = 1/4 s^1/2 m/s^2, where s is in meters. If v = 0, s = 1 m when t = 0, determine the particle’s velocity at s = 2 m.
Answer:
The particle’s velocity is 0.781 m/s.
Explanation:
Given;
acceleration of the particle, [tex]a = \frac{1}{4} s^{\frac{1}{2}} \ m/s^2[/tex] [tex]= 0.25s^{0.5} \ m/s^2[/tex]
Acceleration is given by;
[tex]a = \frac{dv}{dt}\\\\a = \frac{dv}{dt} *\frac{ds}{ds} = \frac{ds}{dt}* \frac{dv}{ds}\\\\a = v*\frac{dv}{ds} \\\\ads = vdv\\\\\int\limits^s_1 {a} \, ds = \int\limits^v_0 {v} \, dv\\\\ \int\limits^s_1 {0.25s^{0.5}} \, ds = \int\limits^v_0 {v} \, dv\\\\\frac{1}{6} (s^{1.5} -1^{1.5}) = \frac{v^2}{2} \\\\v^2 = \frac{2}{6} (s^{1.5} -1^{1.5})\\\\v^2 = \frac{1}{3} (s^{1.5} -1^{1.5})\\\\when \ s= 2 m\\\\v^2 = \frac{1}{3} (2^{1.5} -1^{1.5})\\\\v^2 = 0.6095\\\\v = \sqrt{0.6095}\\\\v = 0.781 \ m/s[/tex]
Therefore, the particle’s velocity at s = 2 m, is 0.781 m/s.
