THE HEIGHT (IN METERS) OF A PROJECTILE SHOT VERTICALLY UPWARD FROM A POINT 5.5 M ABOVE GROUND LEVEL WITH AN INITIAL VELOCITY OF $25.48 M / S$ [tex]h=5.5+25.48 t-4.9 t^2[/tex] IS AFTER T SECONDS.
A) WHEN DOES THE PROJECTILE REACH ITS MAXIMUM HEIGHT?
B) WHAT IS THE MAXIMUM HEIGHT?
Answer (2)
Find the time at maximum height using the vertex formula for a parabola: t = − 2 a b = − 2 ( − 4.9 ) 25.48 = 2.6 seconds.
Substitute t = 2.6 into the height equation: h ( 2.6 ) = 5.5 + 25.48 ( 2.6 ) − 4.9 ( 2.6 ) 2 .
Calculate the maximum height: h ( 2.6 ) = 5.5 + 66.248 − 33.124 = 38.624 meters.
The projectile reaches its maximum height of 38.624 meters at 2.6 seconds.
Explanation
Problem Analysis We are given the height of a projectile as a function of time: h ( t ) = 5.5 + 25.48 t − 4.9 t 2 . We need to find the time when the projectile reaches its maximum height and the maximum height itself.
Finding Time at Maximum Height To find the time when the projectile reaches its maximum height, we need to find the vertex of the parabolic equation. The vertex occurs at t = − 2 a b , where a = − 4.9 and b = 25.48 .
Calculating Time at Maximum Height Substituting the values of a and b , we get t = − 2 ( − 4.9 ) 25.48 = 9.8 25.48 = 2.6 . Therefore, the projectile reaches its maximum height at t = 2.6 seconds.
Finding the Maximum Height To find the maximum height, we substitute the value of t we just found ( t = 2.6 ) into the equation for h ( t ) : h ( 2.6 ) = 5.5 + 25.48 ( 2.6 ) − 4.9 ( 2.6 ) 2 .
Calculating the Maximum Height Value Calculating the maximum height: h ( 2.6 ) = 5.5 + 66.248 − 4.9 ( 6.76 ) = 5.5 + 66.248 − 33.124 = 38.624 . Therefore, the maximum height of the projectile is 38.624 meters.
Final Answer The projectile reaches its maximum height at 2.6 seconds, and the maximum height is 38.624 meters.
Examples
Understanding projectile motion is crucial in sports like basketball or soccer, where players need to estimate the trajectory and maximum height of a ball to make accurate shots or passes. Similarly, in engineering, calculating the trajectory of a rocket or missile is essential for mission planning and ensuring it reaches its intended target. This problem demonstrates how quadratic equations can model real-world phenomena and help predict outcomes.
The projectile reaches its maximum height at approximately 2.6 seconds, and the maximum height is about 38.624 meters.
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