Hey Guys,

Could someone please help with this question, I am really struggling. Any help will be greatly appreciated!

1. Jamie is playing in an AFL game.1 He kicks the ball from position O = (0; 0; 0) towards the goal. The bases of the goalposts are at (40; 10; 0) and (40; 16:4; 0). The ball follows a path described by the parametric equations (x(t); y(t); z(t)) = (10t; 3t; 8ROOT5t - 4t^2) (where t is in seconds), until it hits the ground, at which point we will assume for simplicity that it stops and does not bounce.

(a) At what time does the ball hit the ground?

(b) If the ball passes between the goalposts, what must its x-coordinate be at that moment?

What conditions must the time t and y-coordinate satisfy?

(c) In the absence of any opposing players who might have been able to stop the ball, did Jamie score a goal?

(d) Sketch a graph of the path the ball followed, as seen from above, i.e. in the x-y plane. Indicate the positions of the goalposts, and label the important features of the graph.

Thank you soo much in advance!!

The bases of the goalposts are at (40; 10; 0) and (40; 16.4; 0)

Hey Guys,

...
1. Jamie is playing in an AFL game.1 He kicks the ball from position O = (0; 0; 0) towards the goal. The bases of the goalposts are at (40; 10; 0) and (40; 16:4; 0). The ball follows a path described by the parametric equations (x(t); y(t); z(t)) = (10t; 3t; 8ROOT5t - 4t^2) (where t is in seconds), until it hits the ground, at which point we will assume for simplicity that it stops and does not bounce.

...
Good morning,
(even though it is late afternoon in Melbourne)

I don't understand your equation of the flight path of the ball. Do you mean:

$p(t)=\left \lbrace \begin{array}{l}x(t)=10t \\ y(t) = 3t \\ z(t)=\sqrt[8]{5t-4t^2}\end{array}\right.$

If and only if this is correct the ball flies only 1.25 s, that means it only reaches the point (12.5 , 3.75 , 0). So Jamie should better play in the kindergarden

Hi I have just attached an image of the question.

So the equation of the flight path is actually:

$p(t)=\left \lbrace \begin{array}{l}x(t)=10t \\ y(t) = 3t \\ z(t)=8 \cdot \sqrt{5}t-4t^2\end{array}\right.$

(a) At what time does the ball hit the ground?

(b) If the ball passes between the goalposts, what must its x-coordinate be at that moment?

What conditions must the time t and y-coordinate satisfy?

(c) In the absence of any opposing players who might have been able to stop the ball, did Jamie score a goal?
to #a): When the ball hits the ground z(t) = 0 (No underground football possible in AUS). Solve for t:

$8 \cdot \sqrt{5}t-4t^2 = 0$

to #b): If the ball passes between the posts the following conditions must be satisfied:

$x(t) \ge 40~\wedge~ 10 < y(t) < 16.4$

Solve first for t: $x(t) \ge 40$ and then check if the 2nd condition is satisfied too.

to #c): Combine the answers of a) and b) and check if Jamie's shot is successfull.