Lesson summary
1.1.5 Resolving Vectors — EDEXCEL IAL PHYSICS (IAS & A2) notes: Resolve a vector into perpendicular components by scale drawing and by trigonometry.Representation — an arrow whose head gives the direction and whose length gives the magnitudeResolving — representing a single vecto…
- Independence
- Horizontal component
- Vertical component
- Angle to the vertical
- Recombining
- Specified method
Section 1 of 9Vectors and components
Vectors and components
Representation
- Representation an arrow whose head gives the direction and whose length gives the magnitude
Resolving
- Resolving representing a single vector by two perpendicular vectors that together have the same effect
Components
- Components the parts into which a vector is broken, often drawn dotted with a subscript for horizontal or vertical
Independence
- Independence perpendicular components do not affect each other, which makes projectiles and forces at angles easier
Section 2 of 9Resolving by calculation
Resolving by calculation
Diagram
- Diagram a sketch is enough if the resultant, components and sides are labelled
Horizontal component
- Fx = F cos θ for θ to the horizontal
Vertical component
- Fy = F sin θ
Angle to the vertical
- Angle to the vertical the roles of sin and cos swap
Check
- Check the component along the side next to the angle uses cos
Recombining
- F = √(Fx2 + Fy2) and tan θ = Fy/Fx
Mnemonic
- Mnemonic soh-cah-toa
Section 3 of 9Resolving by scale drawing
1
Choose a scale that fits the page, such as 1 cm = 10 m for a 100 m resultant.
2
Mark the start point and a line for the vertical or horizontal direction given in the question.
3
Measure the angle with a protractor and draw the resultant vector accurately.
4
Draw the right-angled triangle on the resultant to form the components.
5
Measure the sides and convert with the scale.
- Tools
- sharp pencil, ruler and protractor, accurate to 1 mm
Section 4 of 9Choice of method
Specified method
- Specified method if the question says calculate or scale drawing, use that method
Free choice
- Free choice any correct method gives the correct answer
Speed
- Speed calculation is quicker and more accurate once trigonometry is secure
Section 5 of 9Choosing axes: inclined planes
Choosing axes: inclined planes
Axes
- Axes they need not be horizontal and vertical
Slope at angle θ
- Slope at angle θ choose one axis along the slope and one perpendicular to it
Weight components
- Weight components mg sin θ down the slope and mg cos θ into the slope
String on a cart
- String on a cart a string pulling at an angle gives a horizontal part that accelerates the cart and a vertical part that reduces the normal reaction
Section 6 of 9Using components
Using components
Newton's second law
- ΣFx = max and ΣFy = may separately
Velocity
- Velocity a ball kicked at an angle starts with horizontal and vertical speeds from cos and sin
Relative motion
- Relative motion for a swimmer crossing a river, the river-direction and across-river components are treated separately,
- and the crossing time depends only on the across-river component
Replacement
- Replacement components replace the original force and are not extra forces
Section 7 of 9More from the specification
More from the specification
Orthogonal component resolution
- Orthogonal component resolution projecting vectors onto perpendicular coordinate axes using sine and cosine functions provides an analytical means to simplify forces.
Vector Resolution into Components
- Any vector quantity,
- such as velocity or displacement, can be resolved into two independent perpendicular components: horizontal (Ax = A cos θ) and vertical (Ay = A sin θ).
Displacement-Time and Velocity-Time Graphs
- On a displacement-time graph, the gradient represents instantaneous velocity;
- on a velocity-time graph, the gradient represents acceleration,
- and the area under the curve gives displacement.
Sign Conventions for Vectors
- Unlike scalar distance, displacement is a vector that can be positive or negative;
- opposing directions of motion must be assigned opposite signs to ensure correct algebraic summation.
SUVAT Equations of Motion
- For an object accelerating at a constant rate, motion along each resolved direction is calculated independently using the suvat equations: v = u + at, s = ut + ½at², v² = u² + 2as,What the symbols mean: v = final velocity (m s−1) · u = initial velocity (m s−1) · a = acceleration (m s−2) · t = time (s)
- and s = ½(u + v)t.
Vector Resolution into Components
- Any vector quantity,
- such as velocity or displacement, can be resolved into two independent perpendicular components: horizontal (Ax = A cos θ) and vertical (Ay = A sin θ).
Displacement-Time and Velocity-Time Graphs
- On a displacement-time graph, the gradient represents instantaneous velocity;
- on a velocity-time graph, the gradient represents acceleration,
- and the area under the curve gives displacement.
Sign Conventions for Vectors
- Unlike scalar distance, displacement is a vector that can be positive or negative;
- opposing directions of motion must be assigned opposite signs to ensure correct algebraic summation.
SUVAT Equations of Motion
- For an object accelerating at a constant rate, motion along each resolved direction is calculated independently using the suvat equations: v = u + at, s = ut + ½at², v² = u² + 2as,
- and s = ½(u + v)t.
Final velocity
- Velocity at the end, v.
Significant figures
- Digits carrying meaningful information in a number.
suvat equations
- The four equations linking s, u, v, a and t for constant acceleration.
Initial velocity
- Velocity at the start, u.
Displacement-time graph
- A graph of displacement against time whose gradient is velocity.
Velocity-time graph
- A graph of velocity against time whose gradient is acceleration and area is displacement.
Scalar quantity
- A quantity with magnitude only, such as potential
Vector quantity
- A quantity with both magnitude and direction.
Displacement
- Change in position.
Distance
- Total length of path travelled, a scalar.
Constant
- Not changing.
Section 8 of 9Key terms
| Term | Meaning |
|---|---|
| Resolving | Representing a vector by two perpendicular vectors with the same combined effect. |
| Component | One of the perpendicular parts of a vector. |
| Horizontal component | F cos θ for an angle θ to the horizontal. |
| Vertical component | F sin θ for an angle θ to the horizontal. |
Section 9 of 9Exam technique
1
Using sin for the horizontal component when the angle is measured from the horizontal.
2
Leaving a calculator in radian mode.
3
Adding the magnitudes of components instead of using Pythagoras.
4
On a slope, using mg sin θ as the force into the slope — it is the down-slope component.
5
Treating components as extra forces.