The position of particle P at time t is given by {x=t, y=3t-1}
and the position of particle Q at time t is given by {x=4t-(t^2), y=t+1}
a) Sketch both paths as well as possible. Be sure to label the paths with the particles (P and Q) traveling on each of them.
b) Find the two point of intersection of the paths exactly using algebra.
c) Do the particles ever collide? Support your answer (one of {Yes|No}) with a reason
the paths should not be too hard to sketch. Eliminating t, we have
y = 3x-1
x = 4(y-1) - (y-1)^2
http://www.wolframalpha.com/input/?i=plot+y+%3D+3x-1%2C+x+%3D+4%28y-1%29+-+%28y-1%29^2+
now (b) and (c) become easy, though you should probably work on them using the parametric equations; you can verify your answers with the x-y equations.
a) To sketch the paths of particle P and Q, we can plot a graph using their respective position equations.
For particle P:
- x = t
- y = 3t - 1
For particle Q:
- x = 4t - t^2
- y = t + 1
We can start by finding a few points for each path by plugging in different values of t:
Particle P:
- t = 0: x = 0, y = -1 (point A)
- t = 1: x = 1, y = 2 (point B)
- t = -1: x = -1, y = -4 (point C)
Particle Q:
- t = 0: x = 0, y = 1 (point D)
- t = 1: x = 3, y = 2 (point E)
- t = -1: x = 5, y = 0 (point F)
Plotting these points on a graph, we get a rough sketch of the paths of particle P and Q:
^
| P
3 | B /
| /
2 | P Q /
| ./ /
1 |/ E /
|/ /
O------|---|---|-----> x
-1|\ C
| \
-2 | \ Q
D
Note: The graph is not perfectly accurate since we only chose a few points to plot, but it gives a rough idea of the paths of the particles.
b) To find the two points of intersection of the paths, we need to solve the system of equations:
For x coordinates:
t = 4t - t^2
t^2 - 3t = 0
t(t - 3) = 0
t = 0 or t = 3
For y coordinates:
3t - 1 = t + 1
3t - t = 2
t = 1
Therefore, the two points of intersection are (x, y):
- Point 1: (0, -1)
- Point 2: (3, 2)
c) To determine if the particles collide, we need to compare their positions at the two points of intersection.
For Point 1 (0, -1):
Particle P: x = t = 0, y = 3t - 1 = -1
Particle Q: x = 4t - t^2 = 0, y = t + 1 = 0
For Point 2 (3, 2):
Particle P: x = t = 3, y = 3t - 1 = 8
Particle Q: x = 4t - t^2 = 3, y = t + 1 = 4
As we can see, at Point 1, the y-coordinate for particle P and Q are different, so they do not collide at this point. Similarly, at Point 2, the y-coordinate for particle P and Q are different, indicating that they still do not collide.
Therefore, based on the position calculations, the particles do not collide.