Question

A woman walks $250\mathrm{m}$ in the direction ${35}^{\circ }$ east of north, then $170\mathrm{m}$ directly east. (a) Using graphical methods, find her final displacement from the starting point. (b) Compare the magnitude of her displacement with the distance she walked.

Short answer

The final displacement from the starting point is the resultant vector, with a magnitude and direction calculated in Step 2. The direction will specify the angle east of north. The magnitude of her displacement will typically be less than the total distance she walked, as she did not walk in a straight line.

Step-by-step solution

Drawing the Vectors

The woman takes two walks. First, she walks 250 m in a direction 35 degrees east of north (call this vector A). Then, she walks 170 m directly east (call this vector B). Draw these two vectors with their tails at the same point. Make sure vector A makes a ${35}^{\circ }$ angle with the north (the vertical line), and vector B is a horizontal line to the east of vector A.

Calculating the Final Displacement

Now we want to find the result of these two walks, which is a single vector that starts at the beginning of her travels and ends at her final position (this is vector R, the resultant vector). Since we're dealing with vectors, we apply the law of cosines:$$R=\sqrt{(A^2+B^2+2AB\cos \theta )}$$Where A is the magnitude of vector A (250 m), B is the magnitude of vector B (170 m), and the angle $\theta$ is the angle between vector A and vector B (which is 180-35=145 degrees when measured clockwise). Substituting the values into the law of cosines gives the magnitude of the resultant vector. Also compute the direction using the sine rule:$$\alpha =\arcsin \left({\displaystyle \frac{B\sin \theta }{R}}\right)$$where $\alpha$ is the angle that vector R makes with vector A.

Compare with Total Distance Walked

Now compare the magnitude of the resultant vector R with the total distance she walked. The total distance is simply A + B.

Key concepts

Vector Addition

When dealing with movements in different directions, such as in our exercise, vector addition becomes an essential tool. A vector represents not only the magnitude of movement - the distance - but also the specific direction of that movement. Hence, vectors are depicted as arrows, where the length corresponds to the magnitude and the orientation indicates the direction.

To add vectors, you can't simply add their magnitudes because you also have to consider their directions. This is typically done by arranging the vectors tip-to-tail and drawing the resultant vector, which stretches from the start of the first vector to the end of the last vector. This diagram forms a polygon - in the example of our exercise, a triangle since there are two vectors.

The process of adding vectors graphically ensures that both direction and magnitude are taken into account. In more complex scenarios or where precision is key, mathematical methods like the triangle or parallelogram law are employed, which will be further explored in connection with the law of cosines.

Law of Cosines

The law of cosines is extremely useful for calculating the resultant vector when you know the magnitudes of two vectors and the angle between them. Think of it as an extension of the Pythagorean theorem that applies to any type of triangle, not just right-angled ones. The formula for the law of cosines is:$$c^2=a^2+b^2-2ab\times \text{cos}(\theta )$$
where $a$ and $b$ are the lengths of the sides of a triangle, $c$ is the length of the side opposite the angle $\theta$.

In our case, the two walks the woman takes form two sides of a triangle, and the law of cosines helps us determine her final displacement (the third side of the triangle) graphically depicted as the resultant vector. It accounts for both the magnitude and direction of the walks, which are crucial in understanding the physical concept of displacement.

Resultant Vector

The resultant vector is the single vector that combines the effect of two or more vectors. In the context of our exercise, it represents the woman's final displacement from the starting point, considering both distances she walked and the angle between them.

As indicated in the solution steps, the magnitude of this resultant vector is found using the law of cosines, taking into account the direction she walked during each leg of her journey (vector A and vector B). The angle of the resultant vector, provided by the sine rule, gives us the orientation of her displacement relative to her initial direction. It’s essential to recognize that this resultant vector doesn’t just give us a measure of distance; it's a comprehensive value that encapsulates her overall change in position.

In conclusion, the resultant vector indicates not only how far but also in what direction the woman ends up in relation to her starting point. Understanding how to calculate the resultant vector is fundamental for a variety of physics and engineering problems involving forces, velocities, and other vector quantities.