A toy rocket is shot upward into the air from an initial height of 1/2 meter above the ground with an initial velocity of 19.6 meters per second. The formula for the vertical motion of an object is h(t)=1/2at^2+vt+s where the gravitational constant, a, is -9.8 meters per second squared, v is the initial velocity, s is the initial height, and h(t) is the height in meters modeled as a function of time, t. Due to a malfunction, the toy rocket explodes when it reaches maximum height. How high above the ground is the toy rocket when it explodes?

Question

A toy rocket is shot upward into the air from an initial height of 1/2 meter above the ground with an initial velocity of 19.6 meters per second. The formula for the vertical motion of an object is h(t)=1/2at^2+vt+s where the gravitational constant, a, is -9.8 meters per second squared, v is the initial velocity, s is the initial height, and h(t) is the height in meters modeled as a function of time, t. Due to a malfunction, the toy rocket explodes when it reaches maximum height. How high above the ground is the toy rocket when it explodes? ​

Bot GPT 3.5 · Answer

To find the maximum height reached by the toy rocket before it explodes, we need to first find the time it takes for the rocket to reach its maximum height. At the maximum height, the vertical velocity of the rocket is 0 m/s (since the rocket momentarily stops before falling back down). We can use the formula for velocity to find the time it takes for the rocket to reach this point: v = at + v 0 = -9.8t + 19.6 9.8t = 19.6 t = 2 seconds Now that we have found the time it takes for the rocket to reach its maximum height, we can substitute this value back into the formula for height to find the maximum height reached by the rocket: h(2) = 1/2(-9.8)(2)^2 + 19.6(2) + 0.5 h(2) = -19.6 + 39.2 + 0.5 h(2) = 20.1 meters Therefore, the toy rocket is 20.1 meters above the ground when it explodes.