Techniques of Vector Addition (Grade 10 NSC Matric Physical Sciences): Revision Notes

Worked Example: Head-to-tail addition 1

Question: A car breaks down and you and your friend help push-start it. You push with a force of 50 N and your friend pushes with a force of 45 N. What is the resultant force on the car?

Solution:

Step 1: Choose a scale and reference direction

• Let's choose the direction to the right as positive
• Scale: 1 cm = 10 N

Step 2: Draw the first vector

• Draw the 50 N force as a 5 cm arrow pointing right

Step 3: Draw the second vector

• Since both forces are in the same direction, draw the 45 N force (4.5 cm) starting from the head of the first vector

Step 4: Measure the resultant

• Draw the resultant from the tail of the first vector to the head of the second vector
• The resultant measures 9.5 cm, which equals 95 N to the right

Worked Example: Head-to-tail addition 2

Question: A rugby player has two teammates pushing him forwards with forces of 60 N and 90 N respectively, while two opposing players push him backwards with forces of 100 N and 65 N. Find the resultant force.

Solution:

Step 1: Choose scale and reference direction

• Scale: 0.5 cm = 10 N
• Positive direction: to the right

Step 2-5: Draw all vectors head-to-tail

• $F_1 = 60$ N (right): 3 cm arrow
• $F_2 = 90$ N (right): 4.5 cm arrow from head of $F_1$
• $F_3 = 100$ N (left): 5 cm arrow from head of $F_2$
• $F_4 = 65$ N (left): 3.25 cm arrow from head of $F_3$

Step 6: Measure resultant

• The resultant vector measures 0.75 cm = 15 N to the left

Worked Example: Adding vectors algebraically 1

Question: A tennis ball rolls towards a wall 10 m away. After striking the wall, it rolls 2.5 m back along the ground. Calculate the ball's resultant displacement.

Solution:

Step 1: Choose positive direction

• Let's choose towards the wall as positive
• Therefore, away from the wall is negative

Step 2: Define vectors algebraically

• $\vec{x_1} = +10.0$ m (towards wall)
• $\vec{x_2} = -2.5$ m (away from wall)

Step 3: Add the vectors $\vec{x_R} = \vec{x_1} + \vec{x_2} = (+10 \text{ m}) + (-2.5 \text{ m}) = +7.5 \text{ m}$

Step 4: State the result The resultant displacement is 7.5 m towards the wall.

Worked Example: Subtracting vectors algebraically

Question: A tennis ball is thrown horizontally towards a wall at 3 m⋅s⁻¹. After striking the wall, it returns at 2 m⋅s⁻¹. Determine the change in velocity.

Solution:

Step 1: Choose positive direction

• Towards the wall = positive
• Away from wall = negative

Step 2: Define vectors

• $\vec{v_i} = +3$ m⋅s⁻¹ (initial velocity)
• $\vec{v_f} = -2$ m⋅s⁻¹ (final velocity)

Step 3: Calculate change in velocity $\Delta\vec{v} = \vec{v_f} - \vec{v_i} = (-2 \text{ m⋅s}^{-1}) - (+3 \text{ m⋅s}^{-1}) = -5 \text{ m⋅s}^{-1}$

Step 4: State the result The change in velocity is 5 m⋅s⁻¹ away from the wall.

Worked Example: Adding forces algebraically

Question: A man applies a force of 5 N on a crate. The crate pushes back with a force of 2 N. Calculate the resultant force.

Solution:

Step 1: Choose positive direction

• Towards the crate (direction of man's push) = positive

Step 2: Define forces algebraically

• $\vec{F_{man}} = +5$ N
• $\vec{F_{crate}} = -2$ N (opposite to man's force)

Step 3: Calculate resultant $\vec{F_{man}} + \vec{F_{crate}} = (5 \text{ N}) + (-2 \text{ N}) = 7 \text{ N}$

Step 4: State result The resultant force is 7 N towards the crate.