The driver of a car sets the cruise control and ties the steering wheel so that the car travels at a uniform speed of 14m/s in a circle with a diameter of 100m .
Part A
Through what angular distance does the car move in 4.00min ?
C = pi * D = 3.14 * 100 = 314 m. = The
Circumference.
d = 14m/s * 240s = 3360 m
Da = (3360/314) * 2pi Rad. = 21.4pi Rad.
To find the angular distance that the car moves in 4.00 minutes, we first need to find the time it takes for the car to complete one full revolution around the circle.
The distance traveled around a circle in one full revolution is equal to the circumference of the circle. The formula for circumference is given by:
C = π * d
where C is the circumference and d is the diameter.
Given that the diameter of the circle is 100m, we can calculate the circumference:
C = π * 100m = 314.16m (rounded to two decimal places)
Since the car is traveling at a uniform speed, we can equate the distance traveled to the product of the speed and the time:
Distance traveled = Speed * Time
The distance traveled in one full revolution is equal to the circumference of the circle. Therefore, we can rewrite the equation as:
C = Speed * Time
Rearranging the equation to solve for time:
Time = C / Speed
Time = 314.16m / 14m/s = 22.44s (rounded to two decimal places)
Now, we can calculate the angular distance by using the formula:
Angular distance = (2π * Time) / Period
Since the car is moving in a circle, the period is the time it takes to complete one full revolution, which we found to be 22.44 seconds.
Angular distance = (2π * 22.44s) / 22.44s = 2π radians
Therefore, the car moves through an angular distance of 2π radians in 4.00 minutes.
To find the angular distance the car moves, we need to first calculate the circumference of the circle in which the car is traveling.
The diameter of the circle is given as 100m, and we know that the circumference of a circle is calculated using the formula:
Circumference = π * diameter
So, the circumference of the circle is:
Circumference = π * 100m = 314.16m.
Now, we know that the car travels around this circle in 4.00 minutes.
To find the angular distance, we can use the formula:
Angular Distance = Linear Distance / Radius
Since the linear distance is equal to the circumference of the circle, we can plug in the values we have:
Angular Distance = 314.16m / 100m = 3.1416 radians.
Therefore, the car moves through an angular distance of approximately 3.1416 radians in 4.00 minutes.