A speed trap is set up with two pressure-activated strips placed across a highway, 110 m apart. A car is speeding along at 33 m/s, while the speed limit is 21 m/s. At the instant the car activates the first strip, the driver begins slowing down. What minimum deceleration is needed in order that the average speed is within the limit by the time the car crosses the second marker? Show work.
21 = (33+x)/2
42 = 33 + x
x = 9 m/s
a = change in speed / time
time = distance/average speed
time = 110/21 = 5.24 seconds
a = (9-33)/5.24 = -4.58 m/s^2
Thanks!
Well, if the car is speeding along and activates the first strip, it's probably thinking, "Oops, I'm in for some trouble now!" But let's not jump to conclusions just yet.
To find the minimum deceleration needed, we need to determine the time it takes for the car to travel the 110 m distance between the two strips.
We can use the equation: distance = initial velocity * time + 0.5 * acceleration * time^2
In this case, the initial velocity is 33 m/s, the final velocity is 21 m/s, and the distance is given as 110 m.
We can rearrange this equation to find the time: time = (final velocity - initial velocity) / acceleration
Plugging in the values, we get: time = (21 m/s - 33 m/s) / acceleration
Now, we need to calculate the time it takes for the car to travel from the first strip to the second strip. Since the car is slowing down, we know that the final velocity will be 0 m/s when it reaches the second strip. Therefore, the time it takes to travel the 110 m is the same as the time it takes for the car to decelerate from 33 m/s to 0 m/s.
So, time = (0 m/s - 33 m/s) / acceleration
Setting these two time equations equal to each other, we have: (21 m/s - 33 m/s) / acceleration = (0 m/s - 33 m/s) / acceleration
Now, I know what you're thinking, these equations look the same! But wait, we can't just cancel out the acceleration just yet, because that would lead to dividing by zero and we all know that's a big no-no.
Therefore, we can cross multiply the equation to get rid of the denominators: 21 m/s - 33 m/s = 0 m/s - 33 m/s
This simplifies to: -12 m/s = -33 m/s
Now, I know what you're thinking, these equations still look the same! But wait, we can divide both sides by -1 to clear up the negative signs: 12 m/s = 33 m/s
Well, now we have a bit of a problem. It seems like we've run into some sort of mathematical paradox. Either that, or my calculations have gone horribly wrong.
I'm sorry to inform you that I made a mistake in my calculations. It seems that the minimum deceleration needed to ensure the average speed is within the limit cannot be determined with the information provided. I apologize for any confusion caused. Maybe it's time for a new math problem, huh?
To find the minimum deceleration needed, we need to consider the average speed and the distance the car needs to cover.
Given:
- Distance between the two pressure-activated strips: 110 m
- Initial speed of the car: 33 m/s
- Speed limit: 21 m/s
Let's assume the car decelerates at a constant rate, "a" (in m/s^2), from its initial speed to the speed limit.
To find the minimum deceleration, we need to determine the time it takes for the car to travel from the first strip to the second strip.
1. Calculate the time it takes for the car to reach the second strip:
Distance = Initial velocity * Time + (1/2) * Acceleration * Time^2
110 m = 33 m/s * T + (1/2) * a * T^2
2. Calculate the time it takes for the car to decelerate from its initial speed to the speed limit:
21 m/s = 33 m/s + a * T
Simplifying this equation, we get:
-12 m/s = a * T
3. Solve the two equations simultaneously to find the value of T:
-12 m/s = a * T
Then substitute this value in equation 1:
110 m = 33 m/s * T + (1/2) * a * T^2
Now we have a system of equations.
To solve this, we can eliminate "T" by multiplying the second equation with 33 m/s:
-12 * 33 m/s = a * T * 33 m/s
This gives us:
-396 m/s = 33 a * T
Now, substitute this into the first equation:
110 m = -396 m/s + (1/2) * a * T^2
Rearrange and simplify the equation:
0 = (1/2) * a * T^2 + 396 m/s - 110 m
0 = (1/2) * a * T^2 + 286 m
To find the minimum deceleration "a" needed, we need this equation to have real roots. Therefore, the discriminant (b^2 - 4ac) must be greater than or equal to zero.
Discriminant = (0)^2 - 4 * (1/2) * 286 m * a = -572ma
Since the discriminant must be greater than or equal to zero:
-572ma >= 0
Divide both sides by -572m:
a >= 0
Therefore, any deceleration greater than or equal to zero (non-negative) will ensure that the average speed is within the limit by the time the car crosses the second marker.
Hence, the minimum deceleration needed is 0 m/s^2.