A car travels 10 km east, then 5 km west, and finally 3 km south. What is the car's displacement?
Understand the Problem
The question is asking for the displacement of a car after it travels in specific directions. To solve this, we need to calculate the total distance traveled east and west, then combine this with the distance traveled south to determine the resultant displacement.
Answer
The resultant displacement is given by the formula $D_r = \sqrt{(D_{\text{net east}})^2 + (D_s)^2}$.
Answer for screen readers
The resultant displacement is given by the formula: $$ D_r = \sqrt{(D_{\text{net east}})^2 + (D_s)^2} $$
Steps to Solve
- Calculate Total Eastward Distance
Let's denote the distance traveled east as $D_e$. If a car travels east for a certain distance, we add that to our total east distance.
- Calculate Total Westward Distance
Now, denote the distance traveled west as $D_w$. If the car travels west, we take this distance and subtract it from the total east distance to find the net east distance. The formula is: $$ D_{\text{net east}} = D_e - D_w $$
- Calculate Total Southward Distance
If the car also travels south, the distance is denoted as $D_s$. We note this distance separately as it will help us determine the total displacement using the Pythagorean theorem later.
- Determine the Resultant Displacement
The resultant displacement can be calculated using the formula: $$ D_r = \sqrt{(D_{\text{net east}})^2 + (D_s)^2} $$ This gives us the magnitude of the total displacement considering both east-west and south distances.
- Calculate the Direction of Displacement
To find the direction of the displacement, we can use the tangent function: $$ \theta = \tan^{-1}\left(\frac{D_s}{D_{\text{net east}}}\right) $$ This angle will tell us the direction of the displacement relative to the east.
The resultant displacement is given by the formula: $$ D_r = \sqrt{(D_{\text{net east}})^2 + (D_s)^2} $$
More Information
The resultant displacement gives us a straight-line distance from the starting point to the endpoint of the car's travel. This is useful in understanding how far away the car is in a direct path, rather than considering the total distance traveled.
Tips
- Forgetting to subtract the westward distance from the eastward distance, leading to an incorrect net distance.
- Mixing up the definitions of distance and displacement; remember that displacement is the shortest distance between two points.
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