A car is driven east for a distance of 41 km, then north for 21 km, and then in a direction 30° east of north for 25 km. Determine (a) the magnitude (in km) of the car's total displacement from its starting point and (b) the angle (from east) of the car's total displacement measured from its starting direction.
To determine the magnitude of the car's total displacement, we can visualize the movements as vector additions. Let's break down the car's movements into individual vectors.
1. The car moves east for 41 km. This can be represented as a vector pointing east with a magnitude of 41 km.
2. The car then moves north for 21 km. This can be represented as a vector pointing north with a magnitude of 21 km.
3. Finally, the car moves in a direction 30° east of north for 25 km. To represent this, we need to find the components of this vector in the north and east directions. To find the north component, we use sin(30°) = 0.5 since sin(angle) = opposite / hypotenuse. Therefore, the north component is 0.5 * 25 km = 12.5 km. For the east component, we use cos(30°) = 0.866, since cos(angle) = adjacent / hypotenuse. So, the east component is 0.866 * 25 km = 21.65 km. We can now represent this vector as a combination of north (12.5 km) and east (21.65 km) components.
To find the car's total displacement, we add up the individual vectors:
Total north component = 21 km + 12.5 km = 33.5 km
Total east component = 41 km + 21.65 km = 62.65 km
Using these totals, we can now find the magnitude of the car's total displacement using the Pythagorean theorem:
Magnitude = sqrt((Total east component)^2 + (Total north component)^2)
Magnitude = sqrt((62.65 km)^2 + (33.5 km)^2)
Magnitude ≈ sqrt(3928.4225 km^2 + 1122.25 km^2)
Magnitude ≈ sqrt(5050.6725 km^2)
Magnitude ≈ 71.07 km (rounded to two decimal places)
Therefore, the magnitude of the car's total displacement from its starting point is approximately 71.07 km.
To find the angle of the car's total displacement measured from its starting direction, we can use trigonometry, specifically the inverse tangent function (atan2). The angle can be calculated using the following formula:
Angle = atan2(Total north component, Total east component)
Angle = atan2(33.5 km, 62.65 km)
Using a calculator or an online tool that supports inverse tangent, we can find:
Angle ≈ 29.65°
Therefore, the angle of the car's total displacement measured from its starting direction is approximately 29.65° east of due north.