Question

during each cycle, the velocity v (in meters per second) of a robotic welding device is given by v=9t-2/9+t^2, where t is time in seconds. find the expression for the displacement s (in meters) as a function of t if s=0 when t=0.​

Answer 1

Answer:

$d = \dfrac{9t^{2} }{2} - \dfrac{2}{9} t + \dfrac{t^3}{3}$

Step-by-step explanation:

Given the equation of velocity w.r.to time 't':

$v=9t-\dfrac{2}{9}+t^2 ...... (1)$

Formula for Displacement:

$Displacement = \text{velocity} \times \text{time}$

So, if we find integral of velocity w.r.to time, we will get displacement.

$\Rightarrow \text{Displacement}=\int {v} \, dt$

$\Rightarrow \int {v} \, dt = \int ({9t-\dfrac{2}{9}+t^2}) \, dt \\\Rightarrow \int{9t} \, dt - \int{\dfrac{2}{9}} \, dt + \int{t^2} \, dt\\\Rightarrow s=\dfrac{9t^{2} }{2} - \dfrac{2}{9} t + \dfrac{t^3}{3} + C ....... (1)$

Here, C is constant (because it is indefinite integral)

Formula for integration used:

$1.\ \int({A+B}) \, dx = \int {A} \, dx + \int{B} \, dx \\2.\ \int({A-B}) \, dx = \int {A} \, dx - \int{B} \, dx \\3.\ \int{x^{n} } \, dx = \dfrac{x^{n+1}}{n+1}\\4.\ \int{C } \, dx = Cx\ \{\text{C is a constant}\}$

Now, it is given that s = 0, when t = 0.

Putting the values in equation (1):

$0=\dfrac{9\times 0^{2} }{2} - \dfrac{2}{9}\times 0 + \dfrac{0^3}{3} + C\\\Rightarrow C = 0$

So, the equation for displacement becomes:

$s=\dfrac{9t^{2} }{2} - \dfrac{2}{9} t + \dfrac{t^3}{3}$