A rookie quarterback throws a football with an initial upward velocity component of 15.4m/s and a horizontal velocity component of 19.5m/s . Ignore air resistance.

What is the equation for this graph and how do I find it?

but we know t = 19.5/x

should be t = x/19.5 of course

wikipedia has a nice article on trajectory. If you scroll town a bit, you come to the equations in terms of θ, x and t.

To determine the equation for the path of the football, we need to analyze the motion in the vertical and horizontal directions separately.

In the horizontal direction, the motion is uniform, meaning the horizontal velocity component remains constant throughout the trajectory. Therefore, we can describe the horizontal motion using the equation:

x = v_x * t

where:
- x is the horizontal distance traveled,
- v_x is the horizontal velocity component,
- t is the time.

In this case, the horizontal velocity component is given as 19.5 m/s. Therefore, the equation for the horizontal motion is:

x = 19.5 * t

Now, let's focus on the vertical motion. The key force acting on the football is gravity, which causes it to accelerate downward. The vertical motion can be described using the following kinematic equation:

y = v_y * t + (1/2) * a * t^2

where:
- y is the vertical displacement,
- v_y is the initial vertical velocity component,
- a is the acceleration (in this case, acceleration due to gravity, which is approximately -9.8 m/s^2),
- t is the time.

The initial vertical velocity component is given as 15.4 m/s. Therefore, the equation for the vertical motion is:

y = 15.4 * t - (4.9 * t^2)

To find the overall equation for the trajectory of the football, we can combine the horizontal and vertical equations by substituting the expression for 'x' into the equation for 'y'. The resulting equation will describe the path of the football in terms of time 't'.

So, the equation for the graph representing the trajectory of the football would be:

y = 15.4 * t - (4.9 * t^2)

Horizontal, no force, u =constant

so
x = 19.5 t so t = x/19.5

Vertical, f = -9.81 m/s^2 * mass = m a
so
a = acceleration = - 9.81
so
a = -9.81
v = Vi - 9.81 t
v = 15.4 - 9.81 t
then
y = initial height + 15.4 t - 4.9 t^2
call initial height of quarterback hand Hi
y = Hi + 15.4 t - 4.9 t^2
but we know t = 19.5/x
so
y = Hi + 15.4 (x/19.5) - 4.9 (x/19.5)^2
that is a parabola, multiply it out