The pronghorn antelope has been observed to travel at an average speed of more than 55 km/hover a distance of 6.0 km. Suppose the antelope runs a distance of 6.0 km in the following manner: it travels 2.0 km in one direction, then turns 90 degree and runs another 4.0 km. Calculate the magnitude of the resultant displacement and the angle the resultant vector makes with the vector for the initial displacemet.
D^2 = X^2 + Y^2 = 2^2 +4^2 = 4 + 16 = 20,
D = sqrt(20) = 4.47km,
tanA = Y/X = 2/4 = 0.5,
A = 26.6 deg.
To solve this problem, we need to use vector addition to find the resultant displacement and calculate the magnitude of the resultant displacement and the angle it makes with the initial displacement.
1. Draw a diagram: Draw a diagram to represent the initial and final displacements. Label the initial displacement vector as A and the final displacement vector as B.
----> A (2.0 km)
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90° | B (4.0 km)
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2. Calculate the resultant displacement vector: To find the resultant displacement vector, we need to add the initial and final displacement vectors. We can use the Pythagorean theorem to find the magnitude of the resultant displacement vector and trigonometry to find the angle it makes with the initial displacement.
Resultant displacement vector R = A + B
R = √(A^2 + B^2), where A = 2.0 km and B = 4.0 km
R = √(2.0^2 + 4.0^2)
R = √(4.0 + 16.0)
R = √20.0
R ≈ 4.47 km (rounded to two decimal places)
3. Calculate the angle: To find the angle between the resultant vector and the initial displacement vector, we can use trigonometry.
Tan(angle) = Opposite/Adjacent = (B/A)
Tan(angle) = (4.0 km)/(2.0 km)
Tan(angle) = 2.0
angle = arctan(2.0)
angle ≈ 63.4° (rounded to one decimal place)
Therefore, the magnitude of the resultant displacement is approximately 4.47 km, and the angle the resultant vector makes with the initial displacement vector is approximately 63.4 degrees.