Two particles move in the xy-plane. At time t, the position of particle A is given by x(t)=5t−5 and y(t)=2t−k, and the position of particle B is given by x(t)=4t and y(t)=t2−2t−1.(a) If k=−6, do the particles ever collide?(b) Find k so that the two particles are certain to collide.k=(c) At the time the particle collide in (b), which is moving faster?A. particle AB. particle BC. neither particle (they are moving at the same speed)

Accepted Solution

Answer:a. No the particles will never collide. b. The second particle is moving faster. Step-by-step explanation:We can tell they never collide based on the fact that they will never have the same two points. We can tell this because there is only one time in which they will have the same x value. To find this amount of time, set the two x values equal to each other and solve for t. 5t - 5 = 4t-5 = -t5 = tSo we know the x value will only be the same at 5 seconds. Now we can input that value and see if the y values are the same. 2t + 6 = t^2 - 2t - 12(5) + 6 = 5^2 - 2(5) - 110 + 6 = 25 - 10 - 116 = 14 (FALSE)Therefore they do not collide. For the second part of the question, we know that the second one is moving faster based on the fact that there is a squared value in the y formula. This shows that it is moving at an exponential rate, which always changes faster than a linear rate.