Hi,
I found online a nice (and seemed easy) math problem.
Rocket A travel from Mars to Earth in 200 days
Rocket B travel from Earth to Mars in 150 days, but take off 30 days later
When they cross each other, which one is the closet to the earth ?
So they give a “flat” answer, without giving any explanation on how they reach this conclusion.
What would be your simplest Mathematical solution for this ?
Thanks.
Some simplifying assumptions.
Let the distance from Earth to Mars be equal to 1, and assume it does not change. Let the direction from Earth to Mars be the positive direction.
Assume that the rockets travel at a constant velocity.
The displacement of the rockets can be represented with the lines
S_B(t) = (1/150)(t-30) = (1/150)t - (1/5)
S_A(t) = (-1/200)t + 1
Where t is time in days since rocket A took off. Notice rocket A has a negative slope (negative velocity) since it is moving from Mars to Earth. Rocket A has an initial position of 1, since it starts at Mars. Rocket B has a horizontal shift to the right of 30 days, representing it taking off later.
The rockets cross where these lines intersect. So
(1/150)t - (1/5) = (-1/200)t + 1
((1/150)+(1/200))t = 1 + (1/5)
(7/600)t = (6/5)
t = 720/7 ~= 103
So the rockets cross approximately 103 days after rocket A took off. The position at that time is
S_A(720/7) = (-1/200)(720/7) + 1
=(-18/35) + 1 = 17/35 ~= 0.49
So when they cross, they are about 49% of the way from Earth to Mars. Just closer to Earth than Mars.
When they cross each other, which one is the closet to the earth ?
This is why you read the while question before trying to answer it. When the cross, they are both the same distance from Earth.