Question

The position of a particle is given by the equation, s = t^3 + 6t^2 + 9t where t is measured in seconds and s in metres. Find the acceleration at time ( t ). What is the acceleration after 4 seconds.(please don't give uneven answers)​

Answer 1

ANSWER:

• The acceleration at time t = 6t + 12 m/s².

• The acceleration after 4 seconds = 36 m/s².

GIVEN:

• The position of a particle is given by the equation, s = t³ + 6t² + 9t where t is measured in seconds and s in metres.

TO FIND:

• The acceleration at time t.

• The acceleration after 4 seconds.

EXPLANATION:

$\sf \dashrightarrow s = t^3 + 6t^2 + 9t \\ \\ \\ \boxed{\bold{\large{\red{\dfrac{ds}{dt}=velocity}}}}\\ \\ \\ \sf \dashrightarrow\dfrac{ds}{dt}=\dfrac{d}{dt} t^3 + 6t^2 + 9t\\ \\ \\ \sf \dashrightarrow v = 3 {t}^{2} + 12t + 9 \\ \\ \\ \boxed{\bold{\large{\blue{\dfrac{dv}{dt}=acceleration }}}}\\ \\ \\ \sf\dashrightarrow\dfrac{dv}{dt}=\dfrac{d}{dt} 3 {t}^{2} + 12t + 9 \\ \\ \\ \sf\dashrightarrow a=6t + 12\\ \\ \\ \textbf{\underline{Acceleration at time t:}} \\ \sf\dashrightarrow a=6t + 12 \ m {s}^{ - 2} \\ \\ \\\textbf{\underline{Acceleration at 4 s:}} \\ \sf\dashrightarrow a=6(4) + 12\\ \\ \\\sf\dashrightarrow a=24+ 12\\ \\ \\\sf\dashrightarrow a = 36 \ m {s}^{ - 2}$

Hence the acceleration at time t = 6t + 12 m/s² and the acceleration after 4 seconds = 36 m/s².