Question

when an object is dropped on a certain earth - like planet, the distance it falls in t seconds, assuming that air resistance is negligible, is given by s(t)=12t² where s(t) is in feet. suppose that a medics reflex hammer is dropped from a hovering helicopter. find (a) how far the hammer falls in 4 sec, (b) how fast the hammer is traveling 4 sec after being dropped, and (c) the hammers acceleration after it has been falling for 4 sec.
(a) the hammer falls feet in 4 seconds. (simplify your answer.)
(b) the hammer is traveling ft/sec 4 seconds after being dropped. (simplify your answer.)
(c) the hammers acceleration is ft/sec² after it has been falling for 4 seconds. (simplify your answer.)

Explanation:

Step1: Find distance in 4 seconds

Substitute $t = 4$ into $s(t)=12t^{2}$.
\[s(4)=12\times4^{2}=12\times16 = 192\]

Step2: Find velocity function

Velocity $v(t)$ is the derivative of the position - function $s(t)$. Using the power rule $\frac{d}{dt}(at^{n})=nat^{n - 1}$, if $s(t)=12t^{2}$, then $v(t)=s^\prime(t)=24t$.

Step3: Find velocity at $t = 4$

Substitute $t = 4$ into $v(t)$.
\[v(4)=24\times4=96\]

Step4: Find acceleration function

Acceleration $a(t)$ is the derivative of the velocity - function $v(t)$. Since $v(t)=24t$, then $a(t)=v^\prime(t)=24$. The acceleration is constant and does not depend on $t$.

Answer:

(a) 192
(b) 96
(c) 24