A circular racetrack with 250m radius lies in the x-y plane and is centered at the origin. A car rounds the track counterclockwise starting at the point (250m , 0).
Find the total distance traveled after one-quarter lap.
Find the total displacement after one-quarter lap.
Find the total distance traveled after one-half lap.
Find the total displacement after one-half lap.
Find the total distance traveled after one complete lap.
Find the total displacement after one complete lap.
1. s1=(2πR)/4=392.7 m
2. d1=250•sqroot(2)=353.6 m
3. s2=(2πR)/2=785.4 m
4. d2=2R=500 m
5 s3=2πR=1571 m
6. d3=0
To find the total distance traveled and total displacement after a certain distance, we need to understand the concepts of distance and displacement.
Distance: Distance is the total length covered by an object during its motion, irrespective of direction. It is a scalar quantity and is always positive.
Displacement: Displacement is the change in position from the initial point to the final point. It takes into account both distance and direction. Displacement is a vector quantity.
For a circular racetrack with a given radius, we can calculate the total distance and displacement for different fractions of a lap using the formulas:
Total Distance = Circumference of the track × Fraction of a lap
Total Displacement = 2 × Radius × Fraction of a lap
Now, let's calculate the values for each question:
1. Total distance traveled after one-quarter lap:
Circumference of the track = 2π × Radius
Total Distance = (2π × Radius) × (1/4) = 2π × 250m × (1/4) = 125π m
2. Total displacement after one-quarter lap:
Total Displacement = 2 × Radius × (1/4) = 2 × 250m × (1/4) = 125m
3. Total distance traveled after one-half lap:
Total Distance = (2π × Radius) × (1/2) = 2π × 250m × (1/2) = 250π m
4. Total displacement after one-half lap:
Total Displacement = 2 × Radius × (1/2) = 2 × 250m × (1/2) = 250m
5. Total distance traveled after one complete lap:
Total Distance = 2π × Radius = 2π × 250m = 500π m
6. Total displacement after one complete lap:
Total Displacement = 2 × Radius = 2 × 250m = 500m
Remember, when working with circular motion, the distance traveled is always greater than or equal to the displacement, as the distance takes into account the entire path traveled, while the displacement only represents the change in position.