Help me do this Differential equations problem?

Question

x(t) = v1t + p1
y(t) = v2t + p2
z(t) = −1
2 gt2 + v3t + p3 ,
where the initial conditions are: p(0)  (p1, p2, p3) are the coordinates of the initial position of the
object; and p0(0)  (v1, v2, v3) are the coordinates of the initial velocity of the object. Thus given p(0)
and p0(0) the motion of the object p(t) is known for all time from the formula above.

Suppose the object is launched into space from the point (0, 0, 0) with initial velocity (1, 0, 1).
Where and when does the ball reach its highest point? Suppose your friend is standing at the
point (1, 0, 0) and has an object of his own that he wants to throw up into space to make contact
with your object. Find (v1, v2, v3) so that your friend’s object makes contact with your object
exactly when both objects are at their highest points. (Hint: why can you say v2 = 0?) Draw a
schematic picture of both curves.

Answer 1

So for the particular system, the equations for your object are

x(t) = t
y(t) = 0
z(t) = -½gt² + t

To maximize z, set dz/dt = 0 and solve for t. That tells you when the maximum is reached. Plug that t-value into the equations above to find out where your object is at that time.

For the second question, the friend's system is

x(t) = (v1)t + 1
y(t) = 0
z(t) = -½gt² + (v3)t

(I'll leave it to you to explain why we can take v2=0)

Evaluate the friend's system at the t-value you found in the first question (that is, the time when your object has reached its maximum height). At this time, the friend's object is to be at the same place as your object; so set his object's x-coordinate at that time equal to your object's x-coordinate at that time, and similarly for the z-coordinates. That will enable you to determine v1 and v3 for his object.