3. (30 points) a slingshot fires a stone straight upward. it takes 3.25 seconds to reach a height of 35.0…

Answer:

We can use the kinematic equation (h = v_0t-\frac{1}{2}gt^{2}), where (h = 35.0\ m), (t = 3.25\ s) and (g= 9.8\ m/s^{2}), and we need to solve for (v_0).

First, rewrite the equation for (v_0):

[ \begin{align*} h&=v_0t-\frac{1}{2}gt^{2}\ v_0t&=h + \frac{1}{2}gt^{2}\ v_0&=\frac{h+\frac{1}{2}gt^{2}}{t} \end{align*} ]

Substitute (h = 35.0\ m), (t = 3.25\ s) and (g = 9.8\ m/s^{2}) into the formula:

[ \begin{align*} v_0&=\frac{35.0+\frac{1}{2}\times9.8\times(3.25)^{2}}{3.25}\ &=\frac{35.0 + \frac{1}{2}\times9.8\times10.5625}{3.25}\ &=\frac{35.0+51.75625}{3.25}\ &=\frac{86.75625}{3.25}\ &\approx26.7\ m/s \end{align*} ]

Explanation:

Step1: Identify the kinematic - equation

We use (h = v_0t-\frac{1}{2}gt^{2}) for vertical - motion.

Step2: Rearrange the equation for (v_0)

We get (v_0=\frac{h+\frac{1}{2}gt^{2}}{t}) by algebraic manipulation.

Step3: Substitute the values

Substitute (h = 35.0\ m), (t = 3.25\ s) and (g = 9.8\ m/s^{2}) into the formula for (v_0).

Step4: Calculate (v_0)

Perform the arithmetic operations to find (v_0\approx26.7\ m/s).