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)=17t². 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/sec/sec after it has been falling for 4 sec. (simplify your answer.)

Explanation:

Step1: Find distance in 4 sec

Substitute $t = 4$ into $s(t)=17t^{2}$. So $s(4)=17\times4^{2}=17\times16 = 272$.

Step2: Find velocity function

The velocity function $v(t)$ is the derivative of the position - function $s(t)$. Using the power rule, if $s(t)=17t^{2}$, then $v(t)=s^\prime(t)=34t$.

Step3: Find velocity at $t = 4$

Substitute $t = 4$ into $v(t)$. So $v(4)=34\times4 = 136$.

Step4: Find acceleration function

The acceleration function $a(t)$ is the derivative of the velocity - function $v(t)$. Since $v(t)=34t$, then $a(t)=v^\prime(t)=34$. The acceleration is constant and does not depend on $t$.

Answer:

(a) 272
(b) 136
(c) 34