A police car is chasing a suspect. The policemen is at point y=0.6 (north of an intersection) and the suspect is at point x=0.8 (east of the intersection). The police car is traveling at 60mph. Their radar gun reads that the suspect's car is getting away at a rate of 20mph. How fast is the suspect moving in the x-direction?
To solve this problem, we can use the concept of velocity components.
Let's define the velocity of the suspect's car as Vx (in the x-direction) and Vy (in the y-direction).
We know that the policemen is at point (0, 0.6) and the suspect is at point (0.8, 0).
We are given that the policemen's car is traveling at a constant speed of 60 mph, which means its velocity can be expressed as Vpolice = 60 mph.
The radar gun reads that the suspect's car is getting away at a rate of 20 mph, which means the velocity of the suspect's car can be expressed as Vx - Vy = 20 mph.
Since the policemen's car is stationary in the y-direction but traveling in the x-direction, its velocity components become Vy = 0 and Vpolice (in the x-direction).
Using the Pythagorean theorem, we can calculate the magnitude of the suspect's velocity:
V^2 = Vx^2 + Vy^2
V^2 = (Vx)^2 + (0)^2 = (Vx)^2
V = |Vx|
Therefore, we have |Vx| = 20 mph.
So the suspect's car is moving at a speed of 20 mph in the x-direction.
To solve this problem, we can use the concept of relative velocities. Let's assume that the positive y-direction is north and the positive x-direction is east.
The police car is stationary in the y-direction (north-south direction), so its velocity in the y-direction is 0 mph. The suspect's car is moving away from the police car at a rate of 20 mph, which means its velocity in the y-direction is -20 mph.
The police car is moving in the x-direction (east-west direction) at a speed of 60 mph, but we need to find the velocity of the suspect in the x-direction.
We can use the Pythagorean theorem to calculate the suspect's overall velocity:
velocity of police car = 60 mph
velocity of suspect's car in y-direction = -20 mph
By rearranging the Pythagorean theorem, we have:
velocity of suspect's car = √(velocity of police car)^2 + (velocity of suspect's car in y-direction)^2
velocity of suspect's car = √(60 mph)^2 + (-20 mph)^2
velocity of suspect's car = √(3600 mph^2 + 400 mph^2)
velocity of suspect's car = √4000 mph^2
velocity of suspect's car ≈ 63.25 mph
Therefore, the suspect's car is moving at approximately 63.25 mph in the x-direction.