two cars start on opposite sides of a circular track. One car has a speed of 0.015 rad/s;the other car has a speed of 0.012 rad/s. If the cars start p radians apart, calculate the time it takes for the fatser car to catch up with the slower car.
the answers :1050s (17.5 min) but i just can't see how its done, plz help me
the quicker car gains on the slower car at a rate of 0.003 rad/s
if they start pi radians apart, then the time needed is pi/.003 = 1047.19 sec.
That's pretty close to 1050 sec.
Well, it seems like the cars are moving in a circular track, and one car is faster than the other. Let's break it down step by step, using a bit of humor along the way!
First, let's see how much time it takes for the faster car to catch up with the slower car. Since they are moving in a circular track, we need to find out how many radians they need to cover to meet each other.
Let's assume that the slower car starts at position 0 radians and the faster car starts at position p radians. The slower car's speed is 0.015 rad/s, and the faster car's speed is 0.012 rad/s.
Now, let's think about it like a race. The faster car is trying to catch up with the slower car, so it needs to cover the distance between them. This distance is p radians.
Since the faster car is faster by 0.003 rad/s (0.015 - 0.012), it will gain on the slower car by 0.003 rad/s.
To find out how much time it takes for the faster car to cover the distance between them, we can divide the distance by the speed at which the faster car is gaining on the slower car.
So, the time it takes can be calculated as follows:
Time = Distance / Speed
Time = p radians / 0.003 rad/s
Calculating this gives us the time it takes for the faster car to catch up with the slower car.
Now, you mentioned that the answer is 1050 seconds or 17.5 minutes. To verify this, we need to know the specific value of p radians.
To solve this problem, we will use the concept of relative angular velocity. Let's denote the angular velocity of the faster car as ωf and the angular velocity of the slower car as ωs.
The relative angular velocity, ωr, can be defined as the difference between ωf and ωs:
ωr = ωf - ωs
In this case, ωf = 0.015 rad/s and ωs = 0.012 rad/s, so we can calculate ωr as follows:
ωr = 0.015 rad/s - 0.012 rad/s
ωr = 0.003 rad/s
Now, we need to find the time it takes for the faster car to catch up with the slower car. Let's denote this time as t.
In the time t, the slower car would have covered a distance of p radians, as they both started p radians apart.
The faster car, with an angular velocity of ωf, would cover a distance of ωf * t in the same time t.
Since the faster car is catching up with the slower car, the distance covered by both cars would be the same:
p = ωf * t
We can rearrange this equation to solve for time t:
t = p / ωf
Given that p is the initial separation angle (in radians) and ωf is the angular velocity of the faster car, we can substitute the values:
t = p / 0.015 rad/s
To find the time in seconds, we just need to convert the result to seconds by multiplying by 60:
t = (p / 0.015) * 60
Now, let's calculate the time it takes for the faster car to catch up with the slower car.
Suppose the initial separation angle p is given as 2π radians (which is a complete lap around the circular track):
t = (2π / 0.015) * 60
t = (200π / 3) * 60
t ≈ 1050 seconds
Therefore, it takes approximately 1050 seconds (or 17.5 minutes) for the faster car to catch up with the slower car.
To calculate the time it takes for the faster car to catch up with the slower car, we can use the concept of relative motion. Since the cars are on a circular track and are moving at different speeds, we need to consider their angular positions.
Let's assume that the slower car starts at position 0 radians and the faster car starts at position p radians. The slower car has a speed of 0.012 rad/s, which means that its angular position increases by 0.012 radians every second. Similarly, the faster car has a speed of 0.015 rad/s, so its angular position increases by 0.015 radians every second.
Now we need to determine at what time the faster car catches up with the slower car. The faster car will catch up when its angular position is equal to the slower car's angular position. Let's denote this time as t.
The angular position of the slower car after time t would be:
Angular position of slower car = 0 + (0.012 rad/s) * t
Similarly, the angular position of the faster car after time t would be:
Angular position of faster car = p + (0.015 rad/s) * t
Since the faster car catches up with the slower car, their positions must be equal at time t:
0 + (0.012 rad/s) * t = p + (0.015 rad/s) * t
Now we can solve this equation for t. First, let's isolate the variables on different sides of the equation:
(0.015 rad/s) * t - (0.012 rad/s) * t = p - 0
Simplifying the equation:
0.003 rad/s * t = p
Now, divide both sides of the equation by 0.003 rad/s to solve for t:
t = p / 0.003 rad/s
Plugging in the given value of p radians, we get:
t = p / 0.003 rad/s = 1050 seconds
So, it takes 1050 seconds (which is equivalent to 17.5 minutes) for the faster car to catch up with the slower car.