21. The dog's total displacement is given by the vector sum,
(0, 2 km) + (5 km, 0) + (0, -7 km) = (5 km, -5 km)
where (<em>a</em>, <em>b</em>) denotes a vector with <em>a</em> as its <em>x</em>-component and <em>b</em> as its <em>y</em>-component. This displacement has magnitude
√((5 km)² + (-5 km)²) = √50 km ≈ 7.1 km
and points due southwest.
22. The description here is quite confusing. In which direction is Spiderman running? Is he, for instance, in or on the back of the train and running toward the front? Or is he running on the ground and the train is approaching him?
It's hard to tell whether the train's speed is relevant. If both Spiderman and GG are standing on the train, then no. Otherwise, you have to account for relative velocities.
23. Using the same notation as in (21), the whale's resultant displacement <em>t</em> seconds after making the turn is
(0, 15 m/s) (8 s) + (-15 m/s, 0) <em>t</em> = ((-15 m/s) <em>t</em>, 40 m)
The magnitude of this displacement is given to be 180 m, so
√((-15 m/s)² <em>t</em> ² + (40 m)²) = 180 m
Solve for <em>t</em> :
√((225 m²/s²) <em>t</em> ² + 1600 m²) = 180 m
(√((225 m²/s²) <em>t</em> ² + 1600 m²))² = (180 m)²
(225 m²/s²) <em>t</em> ² + 1600 m² = 32,400 m²
(225 m²/s²) <em>t</em> ² = 30,800 m²
<em>t</em> ² = (30,800 m²) / (225 m²/s²)
<em>t</em> ² ≈ 136.889 s²
<em>t</em> = 11.7 s
