A person throws a baseball from a height of 4 feet with an initial vertical velocit...
Check the final answer first, then review the worked steps.
Problem
A person throws a baseball from a height of 4 feet with an initial vertical velocity of 20 feet per second. What polynomial models the height of this ball, in feet? Which polynomial represents the difference in the heights of the baseballs t seconds after they are thrown?
Answer
\(-16t^2 + 20t + 4\)
Step-by-step solution
- Identify the standard form of a projectile motion equation: The height of an object launched vertically can be modeled by the quadratic equation $h(t) = -16t^2 + v_0t + h_0$, where $h(t)$ is the height at time $t$, $-16$ represents half the acceleration due to gravity (in feet per second squared), $v_0$ is the initial vertical velocity, and $h_0$ is the initial height.
- Determine the initial velocity ($v_0$): The problem states that the initial vertical velocity is 20 feet per second. So, $v_0 = 20$.
- Determine the initial height ($h_0$): The problem states that the baseball is thrown from a height of 4 feet. So, $h_0 = 4$.
- Substitute the values into the projectile motion equation: Substitute $v_0 = 20$ and $h_0 = 4$ into the standard equation: $h(t) = -16t^2 + 20t + 4$.
- Address the second question: The problem asks for the polynomial that represents the difference in the heights of the baseballs $t$ seconds after they are thrown. However, the problem only provides information for one baseball (Min's little brother's). To find the difference in heights, we would need the height function for a second baseball. Assuming the question implies that the first baseball's height is represented by the polynomial derived in the first part, and there is no information about a second baseball, we cannot determine the difference in heights. If the question is interpreted as asking for the polynomial that models the height of the first baseball, then that polynomial is $-16t^2 + 20t + 4$. If there were a second baseball with height $h_2(t)$, the difference would be $h(t) - h_2(t)$ or $h_2(t) - h(t)$. Since no information is given for a second baseball, we will assume the question is asking for the polynomial for the first baseball and that the second part of the question is either incomplete or a misunderstanding of the provided information. Therefore, the polynomial representing the height of the first baseball is $-16t^2 + 20t + 4$.