Q. 90 - Calculus

Question

While Carol is walking along a straight north-south forest path, her velocity, in feet per minute, after t minutes is given by the preceding graph at the right. Suppose the “positive” direction is north.

(a) Describe the sort of walk Carol must have taken to have this velocity graph. Be sure your description explains the physical significance of the fact that her velocity is zero at $t = 15$ minutes and the fact that her velocity is negative for the second half of her walk.

(b) Find Carol’s average acceleration over the 30-minute walk. Was her acceleration constant over the duration of her walk? Why or why not?

(c) What was Carol’s average velocity over her entire walk? Why does your answer make sense?

Step-by-Step Solution

Answer

Ans: 

•  Part (a):At Carol has 0 velocity which means the displacement of Carol is 0 at that  instant.
• Part (b): average acceleration over time is $\frac{-40}{3}$
• Part (c):  Carol's average velocity over the entire walk is negative 

Step 1. Given information:

• “positive” direction is north 
• Carol is walking along a straight north-south forest path 

Step 2. Solving part (a):

Here the objective is to explain what type of path Carol has followed to have such a velocity graph. From the graph, it is observed that Carol has taken the direction opposite to the north which means he travels in the South direction that is why his displacement becomes negative and as result, he has a negative velocity.

At $t=15$ Carol has 0 velocity which means the displacement of Carol is 0 at that instant.

Step 3. Solving part (b):

Here the objective is to find the average acceleration of Carol over 30 minutes of time.

When $t=0, v\left(0\right)=200$ and when $t=30, v\left(30\right)=-200$.

Therefore average acceleration over time is:

$$\begin{gathered} \frac{v\left(30\right)-v\left(0\right)}{30-0}=\frac{-200-200}{30} \\ =\frac{-400}{30} \\ =\frac{-40}{3} \end{gathered}$$

Hence the acceleration is also negative and it is constant over time as the velocity function is linear

Step 4. Solving part (c):

Here the objective is to determine Carol's average velocity over her entire walk.

Carol's average velocity over the entire walk is negative. It seems that Carol has slowed down her speed, which is not true as the velocity becomes negative because of the direction she has taken.