Question

The acceleration of a particle traveling along a straight line is a = 1/4m/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 1

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.

Step-by-step explanation:

Given;

acceleration of the particle,
`a = (1)/(4) s^{(1)/(2)} \ m/s^2`
`= 0.25s^(0.5) \ m/s^2`

Acceleration is given by;

`a = (dv)/(dt)\\\\a = (dv)/(dt) *(ds)/(ds) = (ds)/(dt)* (dv)/(ds)\\\\a = v*(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\\\\(1)/(6) (s^(1.5) -1^(1.5)) = (v^2)/(2) \\\\v^2 = (2)/(6) (s^(1.5) -1^(1.5))\\\\v^2 = (1)/(3) (s^(1.5) -1^(1.5))\\\\when \ s= 2 m\\\\v^2 = (1)/(3) (2^(1.5) -1^(1.5))\\\\v^2 = 0.6095\\\\v = √(0.6095)\\\\v = 0.781 \ m/s`

Therefore, the particle’s velocity at s = 2 m, is 0.781 m/s.