Extended Response Q- Need help!

[VCE1996]

Anna's brother Jack is playing in an football match. 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, 8√5t − 4t2) (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
Jack 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.

If someone could explain how to do this that would be fantastic! 

[VCE1996]

.... anyone? 

[Stevensmay]

Can you please confirm is the correct equation?

[VCE1996]

Yes it is! 

[VCE1996]

Sorry for the late reply btw

[Conic]

(a) The ball hits the ground when the vertical component (z component) of the vector is 0, therefore we have



So it hits the ground seconds after it's kicked.

(b) We know both of the goals have x coordinates of 40, so the ball needs to have a x coordinate of 40 for it to be a goal. We also know the ball hits the ground at , so the time needs to be between 0 and . The posts have y components of 10 and 16.4, so the y coordinate must be between 10 and 16.4.

(c) The ball has x=40 when t=4, and 4 is within the range of time values. When t=4, the y coordinate is 3x4=12. This is within the range of y coordinates, so he scores the goal.

(d) We have x=10t, so x/10=t. We also know y=3t, so y=3/10x. We use the range of t to find the range of x and y values, i.e. x is between 0 and and y is between 0 and .

Spoiler

[VCE1996]

Thanks heaps!!