Calculating Range of Projectile

To calculate the range of the projectile, we need to find the x-coordinate where the projectile hits the ground. This is the value of x when y is equal to zero.

Finding the x-coordinate

To find the value of x when y is equal to zero, we need to solve the equation:

0 = 10x - (5/9)x^2

Multiplying both sides by 9, we get:

0 = 90x - 5x^2

Simplifying further, we get:

0 = x(90 - 5x)

So, x = 0 or x = 18.

Since the projectile is launched from the origin, we can ignore the x = 0 solution.

Calculating Range

The range of the projectile is the horizontal distance traveled by the projectile. We can calculate this by subtracting the initial x-coordinate (0) from the final x-coordinate (18).

Therefore, the range of the projectile is:

Range = 18 - 0 = 18 meters.

Explanation

The trajectory of the projectile is given by the equation y = 10x - (5/9)x^2. This is a quadratic equation that represents a parabolic path. The x-coordinate represents the horizontal distance traveled by the projectile, while the y-coordinate represents the vertical distance.

To calculate the range of the projectile, we need to find the x-coordinate where the projectile hits the ground. This is the value of x when y is equal to zero. We can find this value by solving the quadratic equation and taking the positive root.

Once we have found the x-coordinate, we can calculate the range of the projectile by subtracting the initial x-coordinate (0) from the final x-coordinate.

In this case, we assumed the acceleration due to gravity (g) to be 10m/s. This is a reasonable assumption for most practical purposes, although the actual value of g may vary depending on the location and altitude.