A particle of mass 45 kg moves in a straight line such that the force (in Newtons) acting on it at time t (in seconds) is given by
225t^4-90 t^2-225.
If at time t = 0 its velocity, v , is given by v ( 0 ) = 2, and its position x (in m) is given by x ( 0 ) = 5, what is the position of the particle at time t?
F=Ma
a=F/m=225/45 t^4-90/45 t^2-225/45
a=5t^4-2t^2-5
v=int a(t)dt
=t^5-2/3 t^3-5t+C
given at t=O, v=2
C=2
if v=t^5-2/3 t^3-5t+2 then
x=INT v(t)dt
x=1/6 t^6+1/6 t^4 +2.5t^2+2t+C
given x(o)=6 implies then C=5
To find the position of the particle at time t, we need to integrate the expression for velocity with respect to time, and then integrate the result with respect to time again to find the expression for position.
Given that the velocity at time t = 0 is v(0) = 2, we can find the expression for velocity by integrating the force expression with respect to time:
v(t) = ∫(225t^4 - 90t^2 - 225) dt
To integrate, we can use the power rule of integration, which states that the integral of x^n with respect to x is (1 / (n + 1)) * x^(n + 1). Applying this rule to each term:
∫(225t^4 - 90t^2 - 225) dt
= (1/5) * t^5 - (1/3) * t^3 - 225t + C
Where C is the constant of integration. Now, we know that v(0) = 2, so we can substitute this into the expression for velocity to solve for C:
2 = (1/5) * 0^5 - (1/3) * 0^3 - 225 * 0 + C
2 = 0 - 0 - 0 + C
C = 2
Therefore, the expression for velocity is:
v(t) = (1/5) * t^5 - (1/3) * t^3 - 225t + 2
To find the expression for position, we integrate the expression for velocity with respect to time:
x(t) = ∫((1/5) * t^5 - (1/3) * t^3 - 225t + 2) dt
Applying the power rule of integration to each term again:
x(t) = (1/30) * t^6 - (1/12) * t^4 - (225/2) * t^2 + 2t + D
Where D is the constant of integration. Now, we know that x(0) = 5, so we can substitute this into the expression for position to solve for D:
5 = (1/30) * 0^6 - (1/12) * 0^4 - (225/2) * 0^2 + 2 * 0 + D
5 = 0 - 0 - 0 + 0 + D
D = 5
Therefore, the expression for position is:
x(t) = (1/30) * t^6 - (1/12) * t^4 - (225/2) * t^2 + 2t + 5
Now, we can substitute the given value of t into the expression for position to find the position of the particle at time t.