Answer:
The answer is (c) v(t) = t² - cos(t) + 5
Step-by-step explanation:
Given the acceleration a(t) = 2t + sin(t), we can calculate the velocity. Acceleration(a) is the time rate of velocity(v), it is the differential of velocity with respect to time(t) and is given by the equation:
[tex]a(t)=\frac{dv(t)}{dt} \\[/tex]
cross multiplying, we get
Therefore [tex]dv(t)= a(t)dt[/tex]
To calculate v(t) we integrate both sides
therefore [tex]\int\limits {dv(t)} =\int\limits{a(t)} \, dt[/tex], but a(t) = 2t + sin(t)
[tex]\int\limits {dv(t)} =\int\limits{(2t+sin(t)}) \, dt[/tex]
[tex]v(t)= \frac{2t^{2} }{2} - cos(t) +c=t^{2} - cos(t) +c\\\\ v(t)=t^{2} - cos(t) +c\\[/tex]
to get c At t = 0, v(0) = 4
[tex]v(0)=0^{2} - cos(0) + c\\4=-1+c\\c-1+1=4+1\\c=5[/tex]
substituting c = 5 we get
[tex]v(t)=t^{2} - cos(t) +5[/tex]
