ANGULAR SPEED OF A RACE CAR. A race car is driven around a circular track at a constant speed of 180miles per hour. If the diameter of the track is 1/2mile, what is the angular speed of the car? Express your answer in revolutions per hour(which is equivalent to laps per hour)
Please show me the steps to solving this.
Thanks.
speed is distance/time
how far is one lap? 1 revolution
What is the time for one lap? time= distance/speedgiven.
what is the angular speed? revolution/time
check the units to make certain angular speed is in revs/hr.
Well, let's go step by step:
Step 1: Find the distance of one lap (or one revolution) of the circular track. Since the track has a diameter of 1/2 mile, the circumference (distance of one lap) can be calculated using the formula C = πd, where C is the circumference and d is the diameter. In this case, C = π * (1/2) = π/2 miles.
Step 2: Find the time it takes to complete one lap of the track. Time = Distance / Speed given. In this case, the distance is the circumference of the circular track, which is π/2 miles, and the speed given is 180 miles per hour. So, the time for one lap is (π/2) / 180 hours.
Step 3: Calculate the angular speed. Angular speed is defined as the number of revolutions per unit of time. Since one lap is equivalent to one revolution, the angular speed is the same as the number of revolutions completed in a given time. In this case, the time is (π/2) / 180 hours. So, the angular speed is (1 revolution) / [(π/2) / 180 hours].
Step 4: Simplify the angular speed and express it in revolutions per hour. To simplify the expression, we can divide 1 by the fraction (π/2) / 180, which is the same as multiplying 1 by the reciprocal of the fraction. So, the angular speed is 1 * (180 / (π/2)) revolutions per hour.
Step 5: Simplify further. Multiplying 1 by (180 / (π/2)) is the same as multiplying 1 by (180 * (2/π)), which simplifies to (360/π) revolutions per hour.
So, the angular speed of the race car is (360/π) revolutions per hour.
To find the angular speed of the race car, we need to calculate the time for one lap and then divide the number of laps per hour by that time.
1. Calculate the distance for one lap:
Since the diameter of the track is given as 1/2 mile, the circumference (distance) of the circular track is π * diameter = π * 1/2 = π/2 miles.
2. Calculate the time for one lap:
Using the formula speed = distance / time, we can rearrange it to find time = distance / speed.
Given that the speed is 180 miles per hour and the distance for one lap is π/2 miles, the time for one lap is:
time = distance / speed = (π/2) / 180 = π / 360 hours.
3. Calculate the angular speed:
Angular speed is measured in revolutions per hour. Since one lap on the circular track is equivalent to one revolution, the angular speed is equal to the number of revolutions per hour.
So, the angular speed = 1 / (time for one lap) = 1 / (π / 360) = 360 / π revolutions per hour.
Therefore, the angular speed of the race car is 360 / π revolutions per hour.
To find the angular speed of the race car, we need to determine the time it takes for one lap and the distance of one lap.
Step 1: Find the distance of one lap.
The diameter of the track is given as 1/2 mile. Since the diameter is twice the radius of a circle, the radius of the track is 1/4 mile (1/2 mile divided by 2). The distance of one lap is equal to the circumference of the track, which can be calculated using the formula: circumference = 2 * π * radius.
So, the distance of one lap is 2 * π * (1/4) miles.
Step 2: Find the time for one lap.
We are given that the car is driven around the track at a constant speed of 180 miles per hour. Time is equal to distance divided by speed, so the time for one lap is (distance of one lap) / (speed).
The time for one lap is (2 * π * (1/4)) miles / (180 miles per hour).
Step 3: Find the angular speed.
The angular speed is measured in revolutions per hour, which is the number of laps the car completes in one hour. In other words, it is the distance traveled in one hour divided by the distance of one lap.
The angular speed is (1 hour) / [(2 * π * (1/4)) miles / (180 miles per hour)].
Simplifying the expression:
Angular speed = (1 hour) * (180 miles per hour) / [2 * π * (1/4) miles]
Angular speed = 720 / (π/2) revolutions per hour.
Therefore, the angular speed of the race car is 1440/π revolutions per hour or approximately 458.37 revolutions per hour.