Question

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.

Answer 1

Answer:

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