F two objects travel through space along two different curves, it's often important to know whether they will collide. (Will a missile hit its moving target? Will two aircraft collide?) The curves might intersect, but we need to know whether the objects are in the same position at the same time. Suppose the trajectories of two particles are given by the vector functions r1(t) = t2, 13t − 36, t2 r2(t) = 7t − 12, t2, 5t − 4

The two objects will collide with the same position vector for all three components at exactly t = 4 s

Explanation:

For two particles starting out at the same time to collide, their position Vector's at the time of collision must be exactly the same.

So, at the collision point, position vector of object 1 is equated to that of object 2.

r₁ = (t², 13t-36, t²)

r₂ = (7t-12, t², 5t-4)

At he point of collision

t² = 7t - 12

t² - 7t + 12 = 0

t² - 4t - 3t + 12 = 0

t(t - 4) - 3(t - 4) = 0

t = 3s or t = 4s

13t - 36 = t²

t² - 13t + 36 = 0

t² - 4t - 9t + 36 = 0

t(t - 4) - 9(t - 4) = 0

t = 9s or 4s

t² = 5t - 4

t² - 5t + 4 = 0

t² - 4t - t + 4 = 0

t(t - 4) - 1(t - 4) = 0

t = 1s or t = 4s

The three components intersect at other times, but at t = 4s, they all intersect at the same time! Meaning that, at this point the two objects are at the same place with the same position vector at that time.

u=13

v=30

s/t=u+1\2at

=13+431/2

=13+265.5

=278.5

the box does not move because you need to use a force greater than 300 n. you are only using 50 n. it needs to at least be 301 n for the box to move because then it is an unbalanced force and it will move the way that the greater force is pushing it. if you use only 300 n it will be a balanced force and will not move because the force you are exerting on the   box is the same as its force.

hope this !

im sorry if i went too in depth, i am going to include a link to a video on khan academy that will explain this.   hope that you like the video and that it explains more in detail to you.

that is the link.