How Do Projectiles Behave in Parametric 3D Space?

[trolling]

Homework Statement

The base of a 20-meter tower is at the origin; the base of a 20-meter tree is at (0,20,0). The ground is flat & the z-axis points upward. The following parametric equations describe the motion of six projectiles each launched at time t = 0 in seconds. (i refers to x-axis, j refers to y-axis, & k refers to z-axis)

1) r(t) = (20 + t^2)k
2) r(t) = (2(t^2))j + (2(t^2))k
3) r(t) = 20i + 20j + (2-t^2)k
4) r(t) = 2tj + (20-t^2)k
5) r(t) = (20-2t)i + 2tj + (20-t)k
6) r(t) = ti + tj + tk

A) Which projectile hits the top of the tree?
B) Which projectile is NOT launched from somewhere on the tower & hits the tree?

2. The attempt at a solution

At t = 0,
1) r = <0, 0, 20>
2) r = <0, 0, 0>
3) r = <20, 20, 20>
4) r = <0, 0, 20>
5) r = <20, 0, 20>
6) r = <0, 0, 0>

These are just guesses
A) II, B) VI

 

[SammyS]

Homework Statement

The base of a 20-meter tower is at the origin; the base of a 20-meter tree is at (0,20,0). The ground is flat & the z-axis points upward. The following parametric equations describe the motion of six projectiles each launched at time t = 0 in seconds. (i refers to x-axis, j refers to y-axis, & k refers to z-axis)

1) r(t) = (20 + t^2)k
2) r(t) = (2(t^2))j + (2(t^2))k
3) r(t) = 20i + 20j + (2-t^2)k
4) r(t) = 2tj + (20-t^2)k
5) r(t) = (20-2t)i + 2tj + (20-t)k
6) r(t) = ti + tj + tk

A) Which projectile hits the top of the tree?
B) Which projectile is NOT launched from somewhere on the tower & hits the tree?

2. The attempt at a solution

At t = 0,
1) r = <0, 0, 20>
2) r = <0, 0, 0>
3) r = <20, 20, 20>
4) r = <0, 0, 20>
5) r = <20, 0, 20>
6) r = <0, 0, 0>

These are just guesses
A) II, B) VI

Why is A correct? At what value of t does the projectile impact the top of the tree?

For B: What are the coordinates of the tower?