Question

4. Imagine You are a mathematician working for a space exploration company, and you are tasked with designing the trajectory of a rocket. The rocket follows a parabolic path after being launched. To ensure a successful mission, you need to calculate the equation of the parabola that represents the rocket's path. Here's the information you have: The rocket is launched from a platform at ground level, so the vertex of the parabola is at the point (0, 0). The maximum height the rocket reaches is 500 meters, and the horizontal distance of the rocket is 200 meters. Write the equation of the parabola in vertex form, considering the given information, and calculate the stretch factor required for the rocket's trajectory.

please show steps i don't understand this! ​

Answer 1

Explanation:

To find the equation of the parabola representing the rocket's path, we need to use the vertex form of a parabola equation, which is:

y = a(x - h)^2 + k

where (h, k) represents the vertex of the parabola.

Given that the vertex of the parabola is at (0, 0), we can substitute these values into the equation:

y = a(x - 0)^2 + 0

Simplifying, we get:

y = ax^2

To calculate the stretch factor (a) required for the rocket's trajectory, we need to consider the given information that the maximum height reached by the rocket is 500 meters, and the horizontal distance is 200 meters.

Since the vertex of the parabola is at (0, 0), the maximum height reached by the rocket corresponds to the y-coordinate of the vertex. Therefore, the maximum height is given by:

500 = a(0)^2

Simplifying, we find that a = 500.

Thus, the equation of the parabola representing the rocket's path is:

y = 500x^2

In this equation, x represents the horizontal distance traveled by the rocket, and y represents the corresponding height at that distance.

By substituting different values of x into the equation, you can calculate the corresponding heights of the rocket at various distances along its trajectory.

It's important to note that this equation assumes a simplified model and neglects factors such as air resistance and the gravitational force. In reality, designing the trajectory of a rocket is a complex task that requires considering various factors and using more sophisticated mathematical models.