After a football is kicked, it reaches a maximum height of 14 m and it hits the ground 32m from where it was kicked. After a soccer ball is kicked, it reaches a maximum height of 8m and it hits the ground 38m from where it was kicked. The paths of both balls are parabolas.
a) create an equation that represents the path of the football.
b) To the nearest tenth of a metre, determine the horizontal distance that both balls have travelled when they are at the same height.
football for example:
starts at (0,0)
vertex at (16, 14) {get it ?}
ground again at (
(32,0)
a (y - 14) = (x-16)^2 or something maybe?
what is a?
goes through (0,0)
so
-14 a = 16^2 = 256
a = -18.3 approx
so
-18.3 (y-14) = (x-16)^2 = x^2 - 32 x + 256 = -18.3 y + 256
or
-18.3 y = x^2 - 32 x
To create an equation that represents the path of the football, we can assume that the path of the ball follows a parabolic shape. The general form of a parabolic equation is given by:
y = ax^2 + bx + c
We are given that the football reaches a maximum height of 14m and hits the ground 32m from where it was kicked. Let's assign coordinates to two points on the parabola: the starting point and the highest point.
Starting point: (0, 0) (where the ball is kicked)
Highest point: (x, 14)
First, let's find the x-coordinate of the highest point. We know that the y-coordinate is 14, so the equation becomes:
14 = ax^2 + bx + c
Since the highest point is symmetric about the vertical line passing through the starting point, the x-coordinate of the highest point should be half the horizontal distance at which the ball hits the ground. Therefore, x = 32/2 = 16.
Substituting this value into the equation, we have:
14 = a(16)^2 + b(16) + c
14 = 256a + 16b + c ...(1)
Next, let's find the other point on the parabola. Since the ball hits the ground, its y-coordinate will be 0, and the x-coordinate will be the horizontal distance at which the ball hits the ground (32 in this case).
Substituting these values into the equation, we have:
0 = a(32)^2 + b(32) + c
0 = 1024a + 32b + c ...(2)
We now have a system of three equations consisting of Equation (1), Equation (2), and the equation for the path of the parabola: y = ax^2 + bx + c.
To solve this system of equations, we can use a method such as substitution or elimination. However, to keep it concise, I will solve it using elimination.
By subtracting Equation (2) from Equation (1), we can eliminate the 'c' term:
14 - 0 = (256a + 16b + c) - (1024a + 32b + c)
14 = -768a - 16b ...(3)
Now, we have two equations with two variables:
14 = -768a - 16b ...(3)
0 = 1024a + 32b ...(2)
We can solve these equations to find the values of 'a' and 'b.'
Multiplying Equation (2) by 8, we get:
0 = 8192a + 256b ...(4)
Adding Equation (3) and Equation (4), we eliminate the 'b' term:
14 + 0 = -768a - 16b + 8192a + 256b
14 = 7424a + 240b
Now, we can solve this equation for 'a':
7424a = 14 - 240b
a = (14 - 240b)/7424
We have obtained the value of 'a' in terms of 'b.' To find the specific values of 'a' and 'b,' we can substitute this equation into Equation (2) and solve for 'b.'
0 = 1024[(14 - 240b)/7424] + 32b
After simplifying this equation, we can solve for 'b.' Once we have the value of 'b,' we can substitute it back into the equation for 'a' to find its value.
Now, to determine the horizontal distance that both balls have traveled when they are at the same height, we need to find the x-coordinate of the highest point of the soccer ball's path.
Given that the soccer ball reaches a maximum height of 8m and hits the ground 38m from where it was kicked, we can follow a similar process as before to find the equation representing the soccer ball's path and then find the x-coordinate of its highest point.
Once we have the x-coordinates of the highest points of both balls' paths, we can calculate the horizontal distance traveled by either ball at that height by subtracting the initial position of the respective ball from its highest point position.
Finally, take the absolute value of the difference to find the horizontal distance traveled when they are at the same height, to the nearest tenth of a metre.
a) To create an equation that represents the path of the football, we can use the general equation of a parabola:
y = ax^2 + bx + c
Let's consider the given information:
At the highest point (maximum height), the football reaches a height of 14m, so at this point, the y-coordinate is 14.
The football is kicked from the origin, so when it hits the ground, the x-coordinate is 32.
To find the equation, we need to find the values of a, b, and c. We have two points on the parabola: (0,0) and (32,0).
1) Substituting the first point (0,0) into the equation:
0 = a(0)^2 + b(0) + c
0 = 0 + 0 + c
c = 0
2) Substituting the second point (32,0) into the equation:
0 = a(32)^2 + b(32) + 0
0 = a(1024) + b(32)
Simplifying the equation:
a = -b/32
Substituting this value of a back into the equation, we have:
0 = (-b/32)(1024) + b(32)
0 = -32b + 1024b
0 = 992b
b = 0
Therefore, b = 0, and since c = 0, the equation simplifies to:
y = ax^2
Since b = 0, there is no linear term in the equation, meaning the path of the football is symmetric around the y-axis.
So, the equation that represents the path of the football is:
y = ax^2
b) To determine the horizontal distance that both balls have traveled when they are at the same height, we need to find the x-coordinate where the y-coordinate is the same for both balls.
For the soccer ball, the maximum height is 8m, and it hits the ground at x = 38.
Using the equation for the parabola of the soccer ball:
y = ax^2
At the highest point (maximum height), the y-coordinate is 8, so:
8 = a(38)^2
To find the value of a, we can rearrange the equation as:
a = 8 / (38)^2
Using this value of a in the equation for the football's path (y = ax^2):
y = (8 / (38)^2) x^2
We can now set the two equations equal to each other since the balls are at the same height:
ax^2 = (8 / (38)^2) x^2
Simplifying:
a = 8 / (1444)
a = 0.00553
Now we can find the x-coordinate where the y-coordinate is the same for both balls. Let's call it x_same_height:
0.00553 x_same_height^2 = ((8 / (38)^2) x_same_height^2
Dividing both sides by x_same_height^2:
0.00553 = 8 / (1444)
Simplifying:
x_same_height^2 = (8 / (1444)) / 0.00553
x_same_height^2 = 0.9577
Taking the square root of both sides to solve for x_same_height:
x_same_height = sqrt(0.9577)
x_same_height ≈ 0.979
To the nearest tenth of a meter, the horizontal distance that both balls have traveled when they are at the same height is approximately 0.979 meters.