A man walks 400 m [E20°N], then 500 m [N20°W], then 300 m [W31°S], and finally 400 m [S13°E]. Find his resultant displacement.
Answer (2)
The man's resultant displacement after walking in various directions is approximately 72.92 m in the direction E58.86°N. This is calculated by breaking each movement into x and y components, summing them, and using the Pythagorean theorem to find the magnitude and direction. Hence, the final result is 72.92 m [E58.86°N].
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Break down each displacement into x and y components using trigonometry.
Calculate the total x and y components by summing the individual components: x t o t a l = 37.79 m, y t o t a l = 62.37 m.
Determine the magnitude of the resultant displacement using the Pythagorean theorem: ma g ni t u d e = x t o t a l 2 + y t o t a l 2 = 72.92 m.
Calculate the direction of the resultant displacement using the arctangent function: θ = arctan ( x t o t a l y t o t a l ) = 58.8 6 ∘ . The resultant displacement is approximately 72.92 m [ E 58.8 6 ∘ N ] .
Explanation
Problem Analysis We are given a problem where a man walks in four different directions and distances. We need to find his resultant displacement, which means we need to find the magnitude and direction of his overall movement from the starting point.
Component Breakdown First, we break down each walk into its North/South (y-component) and East/West (x-component). We will use trigonometry to find these components. Remember that angles are measured from the positive x-axis (East) counterclockwise.
Walk 1 Components Walk 1: 400 m [E20°N] The angle is 20°. The x-component is x 1 = 400 × cos ( 2 0 ∘ ) = 400 × 0.9397 = 375.88 m. The y-component is y 1 = 400 × sin ( 2 0 ∘ ) = 400 × 0.3420 = 136.81 m.
Walk 2 Components Walk 2: 500 m [N20°W] The angle is 90° + 20° = 110° (or we can think of it as -20° from the y-axis). The x-component is x 2 = 500 × cos ( 11 0 ∘ ) = − 500 × sin ( 2 0 ∘ ) = − 500 × 0.3420 = − 171.01 m. The y-component is y 2 = 500 × sin ( 11 0 ∘ ) = 500 × cos ( 2 0 ∘ ) = 500 × 0.9397 = 469.85 m.
Walk 3 Components Walk 3: 300 m [W31°S] The angle is 180° + 31° = 211°. The x-component is x 3 = 300 × cos ( 21 1 ∘ ) = − 300 × cos ( 3 1 ∘ ) = − 300 × 0.8572 = − 257.16 m. The y-component is y 3 = 300 × sin ( 21 1 ∘ ) = − 300 × sin ( 3 1 ∘ ) = − 300 × 0.5150 = − 154.54 m.
Walk 4 Components Walk 4: 400 m [S13°E] The angle is 360° - 13° = 347° (or -13°). The x-component is x 4 = 400 × cos ( 34 7 ∘ ) = 400 × sin ( 1 3 ∘ ) = 400 × 0.2250 = 90.08 m. The y-component is y 4 = 400 × sin ( 34 7 ∘ ) = − 400 × cos ( 1 3 ∘ ) = − 400 × 0.9744 = − 389.75 m.
Total Displacement Components Now, we add up all the x-components to find the total x-displacement: x t o t a l = x 1 + x 2 + x 3 + x 4 = 375.88 − 171.01 − 257.16 + 90.08 = 37.79 m.
And we add up all the y-components to find the total y-displacement: y t o t a l = y 1 + y 2 + y 3 + y 4 = 136.81 + 469.85 − 154.54 − 389.75 = 62.37 m.
Magnitude of Resultant Displacement To find the magnitude (total distance) of the displacement, we use the Pythagorean theorem: ma g ni t u d e = x t o t a l 2 + y t o t a l 2 = ( 37.79 ) 2 + ( 62.37 ) 2 = 1427.90 + 3890.02 = 5317.92 = 72.92 m.
Direction of Resultant Displacement To find the direction, we use the arctangent function: \theta = \arctan(\frac{{y_{{total}}}}{{x_{{total}}}}}) = \arctan(\frac{{62.37}}{{37.79}}) = \arctan(1.6504) = 58.86^{\circ} .
Since both x and y components are positive, the direction is in the first quadrant (North-East).
Final Answer The resultant displacement is approximately 72.92 m [E58.86°N].
Examples
Understanding resultant displacement is crucial in fields like navigation and sports. For instance, a hiker traversing a winding trail can use this concept to determine their direct distance and direction from the starting point. Similarly, in sports like golf or soccer, calculating the resultant displacement of a ball helps players understand the overall effect of a series of movements, optimizing their strategy and improving performance.
