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. 

By constructing a scale drawing, determine the distance from Lake Coniston to base camp. 

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.

Determine an expression for the time taken t for the plane to fly a distance D due north of its current position in this windy region. 

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. 

Assuming the orientation of the plane does not change, calculate the speed of the wind u in km h^–1. 

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. 

Construct a scale drawing to determine φ. 

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

Construct a scale diagram to determine the weight of the pole W and the magnitude of 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. 

Construct a scale diagram to determine the magnitude of the canoeist's resultant velocity. 

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. 

By calculation, or by constructing a diagram, determine the magnitude of F_R. 

You may wish to use the result: 

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. 

Construct a scale diagram to determine the resultant velocity of the boat. 

A ladder rests against a vertical wall as shown. 

Explain how the image shows that there is no coefficient of static friction between the ladder and the wall. 

Draw a vector on the image to show the direction of the resultant force from the ground exerted on the ladder. Label this vector G.

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

Show that the coefficient of static friction between the ladder and the ground at the point of slipping is 0.53. 

You may wish to use the result: 

The ladder weighs 125 N. 

Calculate the magnitude of vector G. 

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

Find, using a suitably labelled sketch, the vector representing the particle's change in velocity. 

Find, using a suitably labelled sketch, the vector representing the particle's instantaneous change in velocity. 

Use your answer to part (b) to deduce a property of the particle's acceleration.

The diagram shows a skier travelling at constant speed down a slope of 35°

The force labelled P is parallel to the slope. The force labelled Q is perpendicular to the slope. Assume that there is no friction between the skis and the snow.

Draw the force triangle and identify the forces labelled P and Q.

Calculate the magnitude of force Q.

Calculate the magnitude of the force P.

Draw a new force diagram and calculate the new resultant force down the slope on the skier.

A man wants to cross a river in a motorboat. The speed of the motorboat in still water is

7.0 ms^-1. The river is 32 m wide. There is a current in the river whose speed with respect to the shore is 5.0 ms^-1.

Draw a vector triangle to show the velocities of the motorboat in still water, the current and the resultant velocity of the motorboat.

Calculate the resultant speed of the motorboat.

The man aims his boat at point X. 

Determine the distance from X further down the river at which he reaches the shore.

A woman in an identical boat leaves from the same spot as the man but wants to land at point X.

Determine the direction in which she must turn her boat to do this.

A cyclist cycles 15.0 km due east, followed by 23.0 km due north and then another 7.0 km east.

Draw a vector diagram to represent the cyclist’s journey.

Calculate the resultant displacement travelled by the cyclist:

Calculate the angle from where the cyclist started to due north of the cyclist final destination.

The cyclist wants to return home.

What angle clockwise from north must he take to go home directly?

State whether impulse is a scalar or vector quantity and explain why.

Electric charge can either be stated positive or negative.

State whether electric charge is a scalar or vector quantity and explain why.

The diagram shows a uniform beam supported by two light cables, AB and AC, which are attached to a single steel cable from a crane. The beam is stationary and in equilibrium.

Draw the vector triangle for this situation labelling the tension in both cables and the weight of the beam. 

The tension in the cable AB is 9 N and the tension in the cable AC is 12 N.

Calculate the resultant force required in the beam BC to keep the system in equilibrium.