Question

If the velocity of a particle which moves along the S-axis is given by V= 2 - 4t + 5t^3/2 , where t is in seconds and V is in meters per seconds. The particle is at the position So = 3 m, when t = 0 1.) Compute the position S when t= 3 sec 2.) Compute the velocity of the particle when t = 3 sec. 3) Compute the acceleration of the particle when t =3 sec. (note: This is a problem under Rectilinear motion with variable acceleration use our Formula V= dS/dt, a=dV/dt, adA= VdV)

Answer 1

1. The position is the integral of velocity:

$$s(t) = \int_0^t5t^{3/2} - 4t + 2dt = 2t^{5/2} - 2t^2+2t+3$$

where the last term comes from the given condition: the particle is at the position So = 3 m, when t = 0.

At $t = 3s$ have:

$$s(3s) = 2\cdot (3)^{5/2} - 2\cdot 3^2 + 2\cdot 3 + 3 \approx 22m$$

2. $v(3s) = 2 - 4\cdot 3 + 5\cdot 3^{3/2} \approx 16m/s$

3. The accelration is the first derivative of velocity:

$$a(t) = -4 + 7.5t^{1/2} \\ a(3s) = -4 + 7.5\cdot 3^{1/2} \approx 9.0 m/s^2$$

Answer. 1) 22m, 2) 16m/s, 3) 9.0m/s^2.