Practice 3A

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Finding resultant magnitude and direction - by Steven Gaskill, 2001

1. A truck driver attempting to deliver some furniture travels 8 km east, turns around and travels 3 km west, and then travels 12 km east to his destination.

a. What distance has the driver traveled?

Here's what you know, Dx₁= 8 km, Dx₂ = -3 km, and Dx₃= 12 km. Add the absolute value of the three distances to get the total distance traveled, so 8 km + 3 km + 12 km = 23 km.

b. What is the driver's total displacement?

Here's what you know, Dx₁= 8 km, Dx₂= -3 km, and Dx₃ = 12 km. Since all vectors are on the same line (x-axis), add all the magnitudes to get the total displacement: 8 km + -3 km + 12 km = 17 km to the east (since 17 is positive).
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2. While following the directions on a treasure map a pirate walks 45.0 m north, then turns and walks 7.5 m east. What single straight-line displacement could the pirate have taken to reach the treasure?

Here's what you know, Dy = 45.0 m, and Dx = 7.5 m. Use the formula d = (Dx^2 + Dy^2)^0.5 and plug in: d = (7.5^2 + 45.0^2)^0.5, so d = 45.6207189772 m = 46 m (significant digits). Use the formula q = tan^-1( Dy / Dx ) and plug in: q = tan^-1( 45.0 m / 7.5 m ), so q = 80.537677792 deg = 81 deg (significant digits). Yeilding, 46 m at 81 deg north of east.
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3. Emily passes a soccer ball 6.0 m directly across the field to Kara, who then kicks the ball 14.5 m directly down the field to Luisa. What is the ball's total displacement as it travels between Emily and Luisa?

Here's what you know, Dx = 6.0 m, and Dy = -14.5 m. Use the formula d = (Dx^2 + Dy^2)^0.5 and plug in: d = (6.0^2 + (-14.5)^2)^0.5, so d = 15.6923548265 m = 15.7 m (significant digits). Use the formula q = tan^-1( Dy / Dx ) and plug in: q = tan^-1( -14.5 m / 6.0 m ), so q = -67.5205656029 deg upfield from the side. To convert to "to the side of downfield" add 90 deg, so -67.5205656029 deg + 90 deg = 22.4794343971 deg = 22 deg (significant digits). Yeilding, 15.7 m at 22 deg to the side of downfield.
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4. A hummingbird flies 1.2 m along a straight path at a height of 3.4 m above the ground. Upon spotting a flower below, the hummingbird drops directly downward 1.4 m to hover in front of the flower. What is the hummingbird's total displacement?

Here's what you know, Dx = 1.2 m, and Dy = -1.4 m. Use the formula d = (Dx^2 + Dy^2)^0.5 and plug in: d = (1.2^2 + (-1.4)^2)^0.5, so d = 1.84390889146 m = 1.8 m (significant digits). Use the formula q = tan^-1( Dy / Dx ) and plug in: q = tan^-1( -1.4 m / 1.2 m ), so q = -49.398705355 deg up from the horizon. To convert to "down from the horizon" multiply by -1, so - 49.398705355 deg * -1 = 49.398705355 deg = 49 deg (significant digits). Yeilding, 1.8 m at 49 deg down from the horizon.
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