Respuesta :
Answer:
It takes 5.755 seconds for the rocket to reach an altitude of s = 100m.
Explanation:
The velocity equation is the integrative of the acceleration equation.
The position equation is the integrative of the accelaration equation.
We have that:
[tex]a(t) = 6 + 0.02t[/tex]
The s there is wrong, m/s² means that it is a(t), not a(s), since s is in meters and t is in seconds.
The velocity is
[tex]v(t) = \int a(t) \, dt[/tex]
[tex]v(t) = \int (6 + 0.02t) \, dt[/tex]
[tex]v(t) = 6t + 0.01t^{2} + K[/tex]
In which K is the constant of integration. This is the initial velocity. We have that v = 0 when t = 0. So K = 0.
[tex]v(t) = 6t + 0.01t^{2}[/tex]
The position is the integrative of the velocity, so:
[tex]s(t) = \int v(t) \, dt[/tex]
[tex]s(t) = \int (6t + 0.01t^{2}) \, dt[/tex]
[tex]s(t) = 3t^{2} + 0.0033t^{3} + K[/tex]
In which K is the constant of integration. This is the initial position. We have that s = 0 when t = 0. So K = 0.
The equation for the position is:
[tex]s(t) = 3t^{2} + 0.0033t^{3}[/tex]
Determine the time needed for the rocket to reach an altitude of s = 100 m.
This is t when s = 100.
[tex]s(t) = 3t^{2} + 0.0033t^{3}[/tex]
[tex]3t^{2} + 0.0033t^{3} = 100[/tex]
[tex]0.0033t^{3} + 3t^{2} - 100 = 0[/tex]
We only use the positive roots, since the answer is an instant of time.
The answer is t = 5.755.
So it takes 5.755 seconds for the rocket to reach an altitude of s = 100m.
