Final answer:
The height at which the firework explodes can be calculated using kinematic equations for one-dimensional motion with the given initial speed and acceleration due to gravity. The time for the firework to ascend before beginning to fall is found first, which is then used to calculate the maximum height using displacement-timed equations.
Step-by-step explanation:
To find the height at which the firework explodes and the time it will be traveling upward, we use kinematic equations for projectile motion under Earth's gravity. Given an initial speed of 69.5 m/s and an acceleration due to gravity of 6.82 m/s2, these equations can help us determine the maximum height and the time taken to reach this height before the firework starts to fall back down. Since the firework is fired straight up, the problem simplifies to a one-dimensional motion.
The time to reach the maximum height is found by using the equation vf = vi + at, where vf (final velocity) is 0 m/s at the highest point, vi (initial velocity) is 69.5 m/s, and a (acceleration) is -6.82 m/s2 (negative because gravity is working opposite to the direction of motion). From this, the time is calculated as t = -vi/a = -69.5 m/s / -6.82 m/s2, which gives the time it takes for the firework to stop ascending.
To determine the height at which the firework explodes, we use the equation s = vit + 0.5at2, where s is the displacement. Plugging in the values, we can calculate the maximum height. This will be the height at which the firework explodes.