We are given the formula for the position $S(t)$ as a function of time:

$S(t) = \frac{1}{2} a t^2 + v_0 t$

where:

• $a = 0.002 \, \text{miles/second}^2$ (acceleration),
• $v_0$ is the initial velocity (Valerie's starting velocity).

Valerie's initial velocity is $35 \, \text{miles per hour}$. First, we need to convert this to miles per second.

Step 1: Convert $35 \, \text{miles/hour}$ to $\text{miles/second}$

There are 3600 seconds in one hour, so we convert as follows:

$v_0 = 35 \, \text{miles/hour} \times \frac{1 \, \text{hour}}{3600 \, \text{seconds}} = \frac{35}{3600} \, \text{miles/second} \approx 0.009722 \, \text{miles/second}$

Step 2: Substitute $a$ and $v_0$ into the equation for $S(t)$

The given acceleration $a = 0.002 \, \text{miles/second}^2$, and the initial velocity $v_0 = 0.009722 \, \text{miles/second}$. Substituting these values into the formula:

$S(t) = \frac{1}{2} \times 0.002 \times t^2 + 0.009722 \times t$

Simplifying:

$S(t) = 0.001 t^2 + 0.009722 t$

Step 3: Express $S(t)$ in vertex form

The vertex form of a quadratic equation is generally written as:

$S(t) = a(t - h)^2 + k$

where $(h, k)$ is the vertex of the parabola.

To convert $S(t) = 0.001 t^2 + 0.009722 t$ to vertex form, we complete the square.

1. Factor out $0.001$ from the quadratic and linear terms:

$S(t) = 0.001 \left( t^2 + \frac{0.009722}{0.001} t \right) = 0.001 \left( t^2 + 9.722 t \right)$

2. Complete the square:
   • Take half of $9.722$, square it, and add it inside the parentheses.
   • Half of $9.722$ is $\frac{9.722}{2} = 4.861$, and squaring it gives $4.861^2 = 23.625$.

So we add and subtract $23.625$ inside the parentheses:

$S(t) = 0.001 \left( t^2 + 9.722 t + 23.625 - 23.625 \right)$

$S(t) = 0.001 \left( (t + 4.861)^2 - 23.625 \right)$

3. Distribute the $0.001$:

$S(t) = 0.001 (t + 4.861)^2 - 0.023625$

Final Answer:

The position function in vertex form is:

$S(t) = 0.001 (t + 4.861)^2 - 0.023625$

Would you like more details on any part of this calculation, or do you have any other questions?

5 Related Questions:

1. How do you interpret the vertex of the quadratic function in terms of time and position?
2. What does the coefficient $0.001$ in the vertex form represent in real-world terms?
3. How would the equation change if Valerie started from rest instead of 35 mph?
4. How can we calculate the time it takes for Valerie to reach a certain position using this equation?
5. What role does completing the square play in converting to vertex form?

Tip:

Always double-check units when solving physics problems, especially when converting between different units like miles per hour and miles per second.