1.3 Vectors & Scalars

A plane flying across the Lake District sets off from base camp to Lake Windermere, 28 km away, in a direction of 20.0° north of east. 

After dropping off supplies it flies to Lake Coniston, which is 19 km at 30.0° west of north from Lake Windermere. 

The plane now flies due north with a speed v. It moves through air that is stationary relative to it. 

Suddenly, the plane enters a region where the wind is blowing with a speed u from a direction of θ anticlockwise from south.

In still air, the plane travels 180 km every 30 minutes. In the windy region described in part (c), the aircraft takes an extra 4 minutes to travel the same distance, when the wind blows at an angle 53° anticlockwise from south. 

The wind now blows due south with the same speed as in part (c). The plane continues to travel at the same speed in this windy region. 

The pilot wishes to cross the sky along the straight line AB. In order to do so, they must turn the plane at an angle φ clockwise from north. 

Two taut, light ropes keep a pole vertically upright by applying two tension forces, one of magnitude 200 N and one of magnitude T. 

A canoeist can paddle at a speed of 3.8 m s^–1 in still water. But, she encounters an opposing current, moving at a speed of 1.5 m s^–1 at 30° to her original direction of travel. 

The boat shown is being towed at a constant velocity by a towing rope, which exerts a tension force F_T = 2500 N. There are two resistive forces indicated – the force of the water on the keel F_K and the force of the water on the rudder, F_R. 

Another boat wishes to cross a river. The river flows from west to east at a constant velocity of 35 cm s^–1 and the boat leaves the south bank, due north, at 1.5 m s^–1. 

A ladder rests against a vertical wall as shown. 

G acts at an angle of 62° to the ground. 

The ladder weighs 125 N. 

The linear velocity v of a particle moving in a circle is tangential to its orbit.