Respuesta :
Answer:
The particles collide when t = 7 at the point (49, 49, 49).
Step-by-step explanation:
We know the trajectories of the two particles,
[tex]r_1(t)=\langle t^2,16t-63,t^2\rangle\\r_2(t)=\langle 9t-14,t^2,13t-42\rangle[/tex]
To find if the tow particles collide you must:
- Equate the x-components for each particle and solve for t
[tex]t^2=9t-14\\t^2-9t+14=0\\\left(t^2-2t\right)+\left(-7t+14\right)=0\\t\left(t-2\right)-7\left(t-2\right)=0\\\left(t-2\right)\left(t-7\right)=0[/tex]
The solutions to the quadratic equation are:
[tex]t=2,\:t=7[/tex]
- Equate the y-components for each particle and solve for t
[tex]16t-63=t^2\\^2-16t+63=0\\\left(t^2-7t\right)+\left(-9t+63\right)=0\\t\left(t-7\right)-9\left(t-7\right)=0\\\left(t-7\right)\left(t-9\right)=0[/tex]
The solutions to the quadratic equation are:
[tex]t=7,\:t=9[/tex]
- Equate the z-components for each particle and solve for t
[tex]t^2=13t-42\\t^2-13t+42=0\\\left(t^2-6t\right)+\left(-7t+42\right)=0\\t\left(t-6\right)-7\left(t-6\right)=0\\\left(t-6\right)\left(t-7\right)=0[/tex]
The solutions to the quadratic equation are:
[tex]t=6,\:t=7[/tex]
Evaluate the position vectors at the common time. The common solution is when t = 7.
[tex]r_1(7)=\langle 7^2,16(7)-63,7^2\rangle=\langle 49,49,49\rangle\\\\r_2(7)=\langle 9(7)-14,7^2,13(7)-42\rangle=\langle 49,49,49\rangle[/tex]
For two particles to collide, they must be at exactly the same coordinates at exactly the same time.
The particles collide when t = 7 at the point (49, 49, 49).
