A mass hanging from a spring is set in motion and its ensuing velocity is given by for . Assume that the positive
direction is upward and .
a. Determine the position function for .
b. Graph the position function on the interval .
c. At what times does the mass reach its lowest point the first three times?
d. At what times does the mass reach its highest point the first three times?

Ask by Patel Wood. in the United States

Mar 21,2025

Question

A mass hanging from a spring is set in motion and its ensuing velocity is given by for . Assume that the positive
direction is upward and .
a. Determine the position function for .
b. Graph the position function on the interval .
c. At what times does the mass reach its lowest point the first three times?
d. At what times does the mass reach its highest point the first three times?

Ask by Patel Wood. in the United States

Mar 21,2025

UpStudy AI · Accepted Answer

**a. Finding the Position Function**

The velocity is given by  
$$
v(t)=-5\pi\sin(\pi t)
$$
and the initial position is  
$$
s(0)=5.
$$

The position function $$ s(t) $$ is the integral of the velocity:
$$
s(t)= s(0) + \int_0^t v(\tau)\,d\tau.
$$

Substitute the given values:
$$
s(t)= 5 + \int_0^t -5\pi\sin(\pi \tau)\,d\tau.
$$

Factor out the constant:
$$
s(t)= 5 - 5\pi \int_0^t \sin(\pi \tau)\,d\tau.
$$

Now, compute the integral:
$$
\int \sin(\pi \tau)\,d\tau = -\frac{1}{\pi}\cos(\pi \tau).
$$

Thus, evaluating from $$0$$ to $$t$$:
$$
\int_0^t \sin(\pi \tau)\,d\tau = \left[-\frac{1}{\pi}\cos(\pi \tau)\right]_0^t = -\frac{1}{\pi}\cos(\pi t) + \frac{1}{\pi}\cos(0).
$$
Since $$\cos(0)=1$$, this becomes:
$$
\int_0^t \sin(\pi \tau)\,d\tau = -\frac{1}{\pi}\cos(\pi t) + \frac{1}{\pi}.
$$

Substitute this back into the expression for $$ s(t) $$:
$$
s(t)= 5 - 5\pi \left(-\frac{1}{\pi}\cos(\pi t) + \frac{1}{\pi}\right).
$$

Simplify by canceling $$\pi$$:
$$
s(t)= 5 + 5\cos(\pi t) - 5.
$$

This simplifies to:
$$
s(t)= 5\cos(\pi t).
$$

---

**b. Graphing the Position Function**

The position function is  
$$
s(t)=5\cos(\pi t).
$$
Some key features on the interval $$[0,3]$$:
- At $$t=0$$:  
  $$\cos(0)=1$$ so $$s(0)= 5$$.
- At $$t=0.5$$:  
  $$\cos(0.5\pi)=\cos\left(\frac{\pi}{2}\right)=0$$ so $$s(0.5)=0$$.
- At $$t=1$$:  
  $$\cos(\pi)=-1$$ so $$s(1)=-5$$.
- At $$t=1.5$$:  
  $$\cos(1.5\pi)= 0$$ so $$s(1.5)=0$$.
- At $$t=2$$:  
  $$\cos(2\pi)=1$$ so $$s(2)=5$$.
- At $$t=2.5$$:  
  $$\cos(2.5\pi)=0$$ so $$s(2.5)=0$$.
- At $$t=3$$:  
  $$\cos(3\pi)=-1$$ so $$s(3)=-5$$.

The graph is a cosine wave with an amplitude of 5 and a period  
$$
T=\frac{2\pi}{\pi}=2.
$$
It oscillates between 5 (highest point) and $$-5$$ (lowest point), with peaks at $$t=0$$ and $$t=2$$ and troughs at $$t=1$$ and $$t=3$$.

---

**c. Times When the Mass Reaches its Lowest Point**

The mass is at its lowest point when the position $$s(t)$$ attains its minimum value, i.e., when  
$$
\cos(\pi t)=-1.
$$

Solve:
$$
\cos(\pi t)=-1 \quad \Longrightarrow \quad \pi t = \pi + 2\pi k, \quad k \in \mathbb{Z}.
$$

Divide by $$\pi$$:
$$
t = 1 + 2k.
$$

Thus, the first three times (with $$k=0,\,1,\,2$$) are:
$$
t=1,\quad t=3,\quad t=5.
$$

---

**d. Times When the Mass Reaches its Highest Point**

The mass is at its highest point when $$s(t)$$ attains its maximum value, i.e., when  
$$
\cos(\pi t)=1.
$$

Solve:
$$
\cos(\pi t)=1 \quad \Longrightarrow \quad \pi t = 2\pi k, \quad k \in \mathbb{Z}.
$$

Divide by $$\pi$$:
$$
t = 2k.
$$

Thus, the first three times (with $$k=0,\,1,\,2$$) are:
$$
t=0,\quad t=2,\quad t=4.
$$
**a. Position Function:** The position function is $$ s(t) = 5\cos(\pi t) $$. **b. Graphing the Position Function:** The graph is a cosine wave with an amplitude of 5 and a period of 2. It oscillates between 5 (highest point) and -5 (lowest point), with peaks at $$ t = 0 $$ and $$ t = 2 $$, and troughs at $$ t = 1 $$ and $$ t = 3 $$. **c. Times Reaching Lowest Point:** The mass reaches its lowest point at $$ t = 1 $$, $$ t = 3 $$, and $$ t = 5 $$. **d. Times Reaching Highest Point:** The mass reaches its highest point at $$ t = 0 $$, $$ t = 2 $$, and $$ t = 4 $$.

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Extra Insights

a. Determine the position function for .

b. Graph the position function on the interval .

c. At what times does the mass reach its lowest point the first three times?

d. At what times does the mass reach its highest point the first three times?

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