Scalars & Vectors (DP IB Physics: SL): Exam Questions

A small cannon is designed to fire projectiles at an angle of 22° to the horizontal with an initial velocity of v. 

Calculate the vertical component of velocity if v = 10 m s^–1.

State the direction of the horizontal component of velocity. 

State and explain why the horizontal component of velocity stays constant in the absence of air resistance.

There is a point along the projectile's trajectory at which the vertical component of its velocity decreases to zero.

(i) State the location of this point.

[1]

(ii) Explain why the vertical component of the projectile's velocity decreases to zero at this point.

[2]

An object of weight  is at rest on a slope inclined at an angle above the horizontal.

On the diagram, draw the components of the weight along the axes shown. 

Determine an expression for the magnitude of the component of weight acting parallel to the slope. 

Draw and label a vector arrow to represent the normal reaction force acting on the object.

Identify the third force acting on the object and describe its direction with respect to the slope. 

The diagram shows a skier travelling at constant speed down a slope inclined at 35° to the horizontal.

The weight of the skier is . Two other forces, and , act on the skier parallel and perpendicular to the slope, respectively. Assume the friction between the skis and the slope is negligible.

(i) Identify the forces and .

[2]

(ii) Draw and label a vector diagram to represent the three forces acting on the skier.

[2]

Calculate the magnitude of force .

Calculate the magnitude of force .

The diagram shows a straight section of a river where the water is flowing from left to right at a speed of 2.0 m s^-1.

A person crosses the river in a motorboat which moves at a constant speed of 5.0 m s^-1 relative to the water and perpendicular to the current.

Determine, relative to the river bank, the magnitude and direction of the motorboat’s velocity.

The distance between the motorboat's initial starting point and point X is 32 m. After crossing the river, the motorboat reaches point Y.

Calculate the distance between X and Y.

When the motorboat is crossing the river, the motor produces a constant forward force.

Explain why the motorboat moves at a constant speed.

A load W is supported by two strings kept in tension by equal masses m hung from their free ends, with each string passing over a smooth pulley. 

Draw a free body force diagram for the load W, expressing tensional forces in terms of each mass m.  

The mass of the load W is M.

Determine an expression for

(i) m in terms of W, g and the angle to the vertical θ

[2]

(ii) M in terms of m and the angle to the vertical θ.

[1]

A crane hook is held in equilibrium by three forces of magnitude 16.5 kN, T₁ and T₂.

Construct a diagram, including an appropriate scale, to determine the magnitude of T₁ and T₂. 

A crate rests on an inclined plane. 

Describe the effect on X, Y and Z if the angle of inclination increases. 

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 diagram shows that there is no coefficient of static friction between the ladder and the wall. 

Draw a vector arrow on the diagram 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.