The function $s=f(t)$ gives the position of an object moving along the $s$-axis as a function of time $t$. Graph $f$ together with the velocity function $v(t)=\frac{d s}{d t}=r(t)$ and the acceleration function $a(t)=\frac{d^2 s}{d t^2}=r^{\prime \prime}(t)$, then complete parts (a) through (f).
$s=104 t-16 t^2, 0 \leq t \leq 6.5$ (a heavy object fired straight up from Earth's surface at $104 ft / sec$ )
A. The object is at rest when $v(t)=0 ft / sec$, and this occurs at $t=3.25 sec$.
(Type integers or decimals. Use a comma to separate answers as needed.)
B. The object is never at rest.
b. When does it move to down or up? Select the correct answer below, and if necessary, fill in the answer box(es) to complete your choice
(Simplify your answer. Type your answer in interval notation. Use integers or decimals for any numbers in the expression)
A. The object is moving down for $t$ in the interval $\square$ , but is never moving up
B. The object is moving down for $t$ in the interval ( $3.25,6.5$ ] and the object is moving up for $t$ in the interval $[0,3.25$ ).
C. The object is never moving down, but is moving up for $t$ in the interval
$\square$
D. The object is never moving down or up.
c. When does the object change direction? Select the correct answer below, and if necessary, fill in the answer box to complete your choice.
A. The object changes direction at $t =3.25 sec$.
(Type an integer or a decimal. Use a comma to separate answers as needed.)
B. The object never changes direction.
d. When does the object speed up and slow down? Select the correct answer below, and If necessary, fill in the answer box(es) to complete your choice. (Simplify your answer. Type your answer in interval notation. Use integers or decimals for any numbers in the expression.)
A. The object never speeds up, but slows down for $t$ in the interval $\square$ .
B. The object speeds up for $t$ in the interval $\square$ and the object slows down for $t$ in the interval $\square$
$\square$
C. The object speeds up for $t$ in the interval $\square$. bat never slows down.
$\square$
D. The object never speeds up or slows down.
Answer (1)
Find the velocity function by differentiating the position function: v ( t ) = 104 − 32 t .
Find the acceleration function by differentiating the velocity function: a ( t ) = − 32 .
Determine when the object is at rest by setting v ( t ) = 0 , which gives t = 3.25 seconds.
Determine the intervals when the object is moving up ( 0 ≤ t < 3.25 ) and down ( 3.25 < t ≤ 6.5 ).
Conclude that the object changes direction at t = 3.25 seconds, speeds up when 3.25 < t ≤ 6.5 , and slows down when 0 ≤ t < 3.25 .
Explanation
Determining when the object slows down We are given the position function s ( t ) = 104 t − 16 t 2 for an object moving along the s-axis, with 0 \\le t \\[0, 3.25) .
Final Answer Since a ( t ) = − 32 < 0 , the object slows down when 0"> v ( t ) > 0 , which occurs when t < 3.25 . Therefore, the object slows down for t in the interval [ 0 , 3.25 ) .
Examples
Understanding the motion of objects, like a ball thrown in the air, is a classic physics problem. By analyzing the position, velocity, and acceleration functions, we can determine when the ball reaches its highest point, when it's speeding up or slowing down, and how its motion changes over time. This is crucial in fields like sports science, where optimizing performance depends on understanding these factors, or in engineering, when designing systems that involve moving parts and need to control their motion precisely.
