The acceleration of a particle is directly proportional to the square of the time t. When t =0, the particle is at x =24 m. Knowing that at t =6 s, x =96 m and v =18 m/s, express x and v in terms of t.

x(t) = ⅟₁₀₈t⁴ + 10t + 24

v(t) = ⅟₂₇t³ + 10

Explanation:

a(t) = C₁t²

velocity is the integral of acceleration

v(t) = ⅓C₁t³ + C₂

position is the integral of velocity

x(t) = (⅟₁₂C₁)t⁴ + C₂t + C₃

x(0) = 24 = (⅟₁₂C₁)0⁴ + C₂0 + C₃

C₃ = 24

x(6) = 96 = (⅟₁₂C₁)6⁴ + C₂6 + 24

         72 = 108C₁ + 6C₂

         C₂ = 12 - 18C₁

v(6) = 18 = ⅓C₁6³ + C₂

          18 = 72C₁ + C₂

          18 = 72C₁ + (12 - 18C₁)

           6 = 54C₁

           C₁ = 1/9

           C₂ = 12 - 18(1/9)

           C₂ = 10

T = approximately 24 hs.

Explanation:

In order to keep the satellite over a fixed point on the equator, as the earth rotates, the satellite must have the same angular velocity that Earth has, which means that it must have a period equal to the time used by Earth to complete an entire rotation on itself, which is almost exactly 24 Hs.

Mathematically, this can be obtained taking into account that the force that keeps the satellite in orbit is the centripetal force, which is actually the gravitational force exerted by Earth, so we can write the following equality:

Fg = Fc ⇒ G*ms*me / (re +rsat)² = ms*ω²*(re +rsat)

By definition, ω =ΔФ / Δt

For a complete revolution, ΔФ = 2*π, and Δt = T (period of the rotation),

so we can replace ω by (2*π/T), solving then for T:

T= 86,313 sec. (24 hs are exactly 86,400 sec, so the value is actually very close to the theorical one).

Explanation: