A gas station stands at the intersection of a north-south road and an east-west road. A police car is traveling towards the gas station from the east, chasing a stolen truck which is traveling north away from the gas station. The speed of the police car is 100 mph when it is 3 miles from the gas station. At the same time the truck is 4 miles from the gas station going 80 mph. At this moment:
a. Is the distance between the car and the truck increasing or decreasing? How fast?
b. Repeat part a) if the truck is going 70 mph instead of 80 mph
a = distance apart = (x^2 + y^2)^.5
at present a = 5 miles (3,4,5 triangle)
da/dt = .5 (x^2+y^2)^-.5 (2x dx/dt + 2 y dy/dt)
if dx/dt = -100
dy/dt = +80
then
da/dt = (.5/5)(2*3*-100 + 2*4*80)
= (.1)(-600 + 640)
= +4
the truck is making a getaway at the moment.
You can do it for the truck doing 70 mph. I suspect the truck will be losing then.
SO B is -4. which is decreasng
To answer both parts a) and b), we can use the concept of relative motion and the knowledge that the rate at which the distance between two objects changes is given by the difference in their speeds.
Let's answer part a) first:
The police car is traveling east towards the gas station, while the stolen truck is traveling north away from the gas station. Since the car and the truck are moving in different directions, we can subtract their speeds to find the relative speed between them.
Relative speed = Speed of police car - Speed of stolen truck
Relative speed = 100 mph - 80 mph
Relative speed = 20 mph
Now, we know that the distance between the car and the truck is decreasing since the truck is moving away from the gas station and the car is chasing it. The rate at which the distance between them is decreasing is equal to the relative speed, which is 20 mph.
So, the answer to part a) is that the distance between the car and the truck is decreasing at a rate of 20 mph.
Now, let's move on to part b):
If the truck is going 70 mph instead of 80 mph, we need to recalculate the relative speed.
Relative speed = Speed of police car - Speed of stolen truck
Relative speed = 100 mph - 70 mph
Relative speed = 30 mph
Again, we can see that the car is chasing the truck, so the distance between them is decreasing. The rate at which the distance between them is decreasing is given by the relative speed, which is 30 mph.
So, the answer to part b) is that the distance between the car and the truck is decreasing at a rate of 30 mph.
To solve this problem, we can use the concept of relative motion. Let's break down the problem into two parts:
a) We want to determine if the distance between the police car and the truck is increasing or decreasing and how fast.
To do this, we need to find the rate at which the distance between the car and the truck is changing. This can be done by calculating the relative velocity between the two vehicles.
At the moment in question, the police car is 3 miles from the gas station, while the truck is 4 miles from the gas station. The police car is going east, and the truck is going north.
The velocity of the police car is given as 100 mph towards the gas station. The truck's velocity is given as 80 mph.
To find the relative velocity, we need to subtract the velocity of the truck from the velocity of the police car.
Relative velocity = Velocity of police car - Velocity of truck
In this case, the police car is going east, and the truck is going north, so we need to use vector subtraction to calculate the relative velocity.
The magnitude of the relative velocity vector can be found using the Pythagorean theorem:
Relative velocity magnitude = √((Velocity of police car)^2 + (Velocity of truck)^2)
Substituting the given values:
Relative velocity magnitude = √((100 mph)^2 + (80 mph)^2)
Calculating this gives us: Relative velocity magnitude ≈ 128.06 mph
Since the police car is chasing the truck, the relative velocity is negative, indicating that the distance between the two vehicles is decreasing.
Therefore, the distance between the car and the truck is decreasing, and it is decreasing at a rate of approximately 128.06 mph.
b) To repeat part a) with the truck going at 70 mph instead of 80 mph, we can use the same approach and calculate the relative velocity.
Relative velocity magnitude = √((100 mph)^2 + (70 mph)^2)
Calculating this gives us: Relative velocity magnitude ≈ 119.69 mph
Again, the relative velocity is negative, indicating that the distance between the car and the truck is decreasing.
Therefore, the distance between the car and the truck is decreasing, and it is decreasing at a rate of approximately 119.69 mph.