Question

At a fireworks display celebration, a firework rocket is launched upward from the ground with an initial velocity of 160 feet per second. Spectators have watched and wondered how high the rocket will go before it begins to descend back to the ground. The formula for vertical motion is h(t) = 0.5at + vt + s, where the gravitational constant, a, is -32 feet per second, v is the initial velocity, and s is the initial height. Time is measured in seconds, and height h is measured in feet.

Answer 1

The formula for vertical motion, h(t) = 0.5at + vt + s, describes the height (h) of a firework rocket at any given time (t). The gravitational constant (a) is typically taken as -32 feet per second, the initial velocity (v) is 160 feet per second, and the initial height (s) is the height from which the rocket is launched.

Step-by-step explanation:

The given formula represents the vertical motion of the firework rocket as a function of time. Breaking down the components:

h(t): This is the height of the rocket at any time t.

0.5at²: Represents the effect of gravity on the rocket's height. The term is negative (-32t²) because gravity acts downward.

vt: Accounts for the initial upward velocity of the rocket.

s: Represents the initial height from which the rocket is launched.

In this case, the initial velocity (v) is 160 feet per second, and the gravitational constant (a) is -32 feet per second. If the rocket is launched from the ground (s = 0), the formula simplifies to h(t) = -16t² + 160t.

To find the maximum height reached by the rocket, we look for the vertex of the parabolic function. The time t at the vertex is given by t = -b/2a, where b is the coefficient of the linear term. In this case, b = 160 and a = -16. Plugging in these values, we can calculate the time at which the rocket reaches its highest point. Once we have the time, we can plug it back into the original equation to find the maximum height.