Answer:
The velocity of the particle is given by [tex]v(t) = 2\cdot t^{2}-30\cdot t + 3[/tex], where [tex]v[/tex] is in meters per second and [tex]t[/tex] is in seconds.
The displacement of the particle is given by [tex]s(t) = \frac{2}{3}\cdot t^{3}-15\cdot t^{2}+3\cdot t -5[/tex], where [tex]s[/tex] is in meters and [tex]t[/tex] is in seconds.
Explanation:
The acceleration of the particle is given by [tex]a(t) = 4\cdot t -30[/tex], the expression for velocities are obtained by integration:
Velocity
[tex]v(t) = \int {a(t)} \, dt[/tex]
[tex]v(t) = \int {4\cdot t-30} \, dt[/tex]
[tex]v(t) = 4\int {t} \, dt -30\int \, dt[/tex]
[tex]v(t) = 2\cdot t^{2}-30\cdot t + v_{o}[/tex]
Where [tex]v_{o}[/tex] is the initial velocity of the particle, measured in meters per second.
If [tex]t = 0[/tex] and [tex]v(0) = 3[/tex], then:
[tex]3 = 2\cdot (0)^{2}-30\cdot (0)+v_{o}[/tex]
[tex]v_{o} = 3[/tex]
The velocity of the particle is given by [tex]v(t) = 2\cdot t^{2}-30\cdot t + 3[/tex], where [tex]v[/tex] is in meters per second and [tex]t[/tex] is in seconds.
Displacement
[tex]s(t) = \int {v(t)} \, dt[/tex]
[tex]s(t) = \int {2\cdot t^{2}-30\cdot t+3} \, dt[/tex]
[tex]s(t) = 2\int {t^{2}} \, dt - 30\int {t} \, dt +3\int \, dt[/tex]
[tex]s(t) = \frac{2}{3}\cdot t^{3}-15\cdot t^{2}+3\cdot t +s_{o}[/tex]
If [tex]t = 0[/tex] and [tex]s(0) = -5[/tex], then:
[tex]-5 = \frac{2}{3}\cdot (0)^{3}-15\cdot (0)^{2}+3\cdot (0)+s_{o}[/tex]
[tex]s_{o} = -5[/tex]
The displacement of the particle is given by [tex]s(t) = \frac{2}{3}\cdot t^{3}-15\cdot t^{2}+3\cdot t -5[/tex], where [tex]s[/tex] is in meters and [tex]t[/tex] is in seconds.
