Questions: The height of a body moving vertically is given by s = -(1/2) g t^2 + v0 t + s0, g > 0, with s in meters and t in seconds. Find the body's maximum height. Which of the following expressions gives the correct maximum height of the body? (v0)/(2 g) + s0 (v0^2)/g + s0 (v0^2)/(2 g) + s0 (v0)/g + s0

The height of a body moving vertically is given by s = -(1/2) g t^2 + v0 t + s0, g > 0, with s in meters and t in seconds. Find the body's maximum height.

Which of the following expressions gives the correct maximum height of the body?
(v0)/(2 g) + s0
(v0^2)/g + s0
(v0^2)/(2 g) + s0
(v0)/g + s0

Transcript text: The height of a body moving vertically is given by $s=-\frac{1}{2} g t^{2}+v_{0} t+s_{0}, g>0$, with $s$ in meters and $t$ in seconds. Find the body's maximum height. Which of the following expressions gives the correct maximum height of the body? $\frac{v_{0}}{2 g}+s_{0}$ $\frac{v_{0}^{2}}{g}+s_{0}$ $\frac{v_{0}^{2}}{2 g}+s_{0}$ $\frac{v_{0}}{g}+s_{0}$

Solution

Solution Steps

Step 1: Understand the Problem

The problem involves finding the maximum height of a body moving vertically under the influence of gravity. The height $ s $ as a function of time $ t $ is given by the equation:

\[ s = -\frac{1}{2} g t^2 + v_0 t + s_0 \]

where $ g $ is the acceleration due to gravity, $ v_0 $ is the initial velocity, and $ s_0 $ is the initial height.

Step 2: Find the Time at Maximum Height

To find the maximum height, we need to determine when the velocity of the body becomes zero. The velocity $ v $ is the derivative of the height function $ s(t) $:

\[ v = \frac{ds}{dt} = -g t + v_0 \]

Set the velocity to zero to find the time $ t $ at which the maximum height occurs:

\[ -g t + v_0 = 0 \implies t = \frac{v_0}{g} \]

Step 3: Calculate the Maximum Height

Substitute $ t = \frac{v_0}{g} $ back into the height equation to find the maximum height:

\[ s_{\text{max}} = -\frac{1}{2} g \left(\frac{v_0}{g}\right)^2 + v_0 \left(\frac{v_0}{g}\right) + s_0 \]

Simplify the expression:

\[ s_{\text{max}} = -\frac{1}{2} \frac{v_0^2}{g} + \frac{v_0^2}{g} + s_0 \]

\[ s_{\text{max}} = \frac{v_0^2}{2g} + s_0 \]