Two automobiles are traveling on intersecting roads. The first automobile is traveling northeast at 35 km/h. The second automobile begins 7 km north of the first, and it is traveling east at 25 km/h. Assuming that these cars will continue at this rate, will they collide? Assign vector equations to each line and assume that the first automobile begins at the origin.
To determine whether the two automobiles will collide, we need to analyze their positions at any given time.
Let's assign a coordinate system with the first automobile starting at the origin, i.e., (0, 0). The first automobile is traveling northeast, which can be represented as a vector with components (35, 35) km/h. The second automobile begins 7 km north of the first, so its initial position is (0, 7). It is traveling east, which can be represented as a vector with components (25, 0) km/h.
Using vector equations, we can express the positions of the two automobiles as functions of time (t).
The position of the first automobile at time t will be given by:
r1(t) = (35t, 35t) km
The position of the second automobile at time t will be given by:
r2(t) = (25t, 7) km
Now, we want to find out if there exists a t for which the two automobiles have the same position. In other words, we want to find the values of t that satisfy the equation r1(t) = r2(t).
Setting the x-components equal, we get:
35t = 25t
Simplifying, we find:
10t = 0
Since t = 0 is the only solution, it means that the two automobiles will never be at the same position simultaneously.
Therefore, the two automobiles will not collide.
To determine if the two automobiles will collide, we need to find the equations of their paths and see if they intersect.
Let's assign the position of the first automobile as vector A(t) = (x1(t), y1(t)), where x1(t) and y1(t) are its coordinates at time t.
Since the first automobile is traveling northeast, its velocity is given as: v1 = 35 km/h at an angle of 45 degrees with the positive x-axis.
The components of the velocity vector v1 are:
vx1 = v1 * cos(45 degrees) = 35 * cos(45 degrees) = 24.75 km/h
vy1 = v1 * sin(45 degrees) = 35 * sin(45 degrees) = 24.75 km/h
The position vector A(t) can be written as:
A(t) = (x1(t), y1(t)) = (24.75t, 24.75t)
Now, let's assign the position of the second automobile as vector B(t) = (x2(t), y2(t)), where x2(t) and y2(t) are its coordinates at time t.
Since the second automobile is traveling east, its velocity is given as: v2 = 25 km/h at a 90-degree angle with the positive x-axis.
The components of the velocity vector v2 are:
vx2 = v2 * cos(90 degrees) = 25 * cos(90 degrees) = 0 km/h
vy2 = v2 * sin(90 degrees) = 25 * sin(90 degrees) = 25 km/h
The position vector B(t) can be written as:
B(t) = (x2(t), y2(t)) = (0 + 0t, 7 + 25t) = (0, 7 + 25t)
To check if the two automobiles will collide, we need to find the values of t where the x-coordinates and y-coordinates are equal.
Setting x1(t) = x2(t) gives us:
24.75t = 0
Solving this equation, we find t = 0.
Setting y1(t) = y2(t) gives us:
24.75t = 7 + 25t
Solving this equation, we find t = -0.466.
Since t = 0 and t = -0.466 do not satisfy the equation, the two automobiles will not collide.
Therefore, the vector equations for the paths of the first and second automobiles are:
A(t) = (24.75t, 24.75t)
B(t) = (0, 7 + 25t)