1. Earth is moving on the path vector r(t) = t^2t+ 8vectorj+ (5t-6)vector k. Sim

Question

1. Earth is moving on the path vector r(t) = t^2t+ 8vectorj+ (5t-6)vector k. Simultaneously, a meteor is moving along vector q(t) = (10-3t)vector i+2t^2 vector j+ t^2vector k. In both cases, t is measured in unspecified units of time. a) When will the two objects collide, and where? b) Which object will be moving faster when they collide? c) At what angle will they collide? d) Use 3D Calc Plotter to graph the two paths together on the same set of 3D axes. On your graph, mark the point of collision and also indicate the direction that each object is traveling along its respective path. Print out your graph to turn in.

Explanation / Answer

(a):

when r = q

then they will collide

so compairing i components of r and q we get

t^2 + 3t -10 = 0

t = -5 , 2 ......(1)

compairing j component of r and q we get

2t^2 = 8

t = -2, 2.........(2)

compairing k components of r and q we get

t^2 - 5t + 6 = 0

t = 3, 2 ........(2)

the common solution of (1) and (2) and (3) is 2

therefore they collide at t = 2 unit.

(c):

(dr/dt) = 2t i + 0j + 5k

(dq/dt) = -3 i + 4t j + 2t k

at t = 2

dr/dt) = 4 i + 0j + 5k

mod (dr/dt) = sqrt(41)

(dq/dt) = -3 i + 8 j + 4 k

mod(dq/dt) = sqrt(89)

(b): therefore meteor moves faster when they collide.

so they collide at an angle of let theta

cos(theta) =[ (dr/dt).(dq/dt) ] / [mod(dr/dt)*mod(dq/dt) ]

theta = 82.391698766421784100552558378406 degree

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