Mechanics
Vectors and Projectiles
Pearson Edexcel International A Level Physics
Scalars and vectors
- A vector has magnitude and direction; a scalar has magnitude only.
- Displacement is a straight line from start to end, whereas distance is the length of the path.
| Scalar | Vector |
|---|---|
| distance | displacement |
| speed | velocity |
| mass | weight |
| energy, work done, power | force |
| time, temperature, density | acceleration, momentum |
Adding vectors
Two vectors at right angles
resultant of two perpendicular vectorsR = √(F12 + F22)
angle of the resultant to F1tan θ = F2 / F1
- Find the magnitude by Pythagoras and the angle by tan θ.
- State the angle to a named direction, such as the horizontal.
Vectors at any angle
Method: scale drawing
- Choose a scale that makes the diagram large, and write the scale on the diagram.
- Draw the first vector to scale in its direction.
- Draw the next vector from the tip of the first, nose to tail, to make a vector triangle, with all three arrows in correct relative directions and at least two sides in the triangle labelled.
- The resultant runs from the tail of the first vector to the tip of the last.
- Measure its length and use the scale to convert it to a magnitude.
- Measure its angle to horizontal with a protractor and mark it on the diagram.
Measure the resultant from the drawing. Never give a resultant calculated instead of measured from the scale drawing.
Resolving vectors
- A force F at angle θ to a direction has a component F cos θ along that direction and F sin θ at right angles to it.
- Resolve every force horizontally and vertically and treat each direction separately.
- A body hanging in equilibrium from a rope at tension T: the vertical component of T is equal to W.
On a slope
- θ is the angle of the slope to the horizontal.
- Component of weight down the slope: mg sin θ. Component at right angles to the slope: mg cos θ.
Projectiles
- The vertical motion is independent of the horizontal motion: the same time links them.
- Horizontally: no force acts, so a = 0 and the horizontal component of velocity is constant: s = ut.
- Vertically: the acceleration is g downwards, whatever the horizontal velocity. With up positive, a = −g.
- Two objects launched horizontally from the same height land together: the vertical acceleration is the same for both.
- Assume air resistance is negligible.
- Thrown horizontally faster from the same height: the horizontal velocity is larger, while the vertical velocity as the ball hits the ground is not affected. The angle to the horizontal at impact: tan θ = vV/vH so θ decreases.
Method: a projectile launched at angle θ to the horizontal
- Resolve the launch velocity u: horizontal component u cos θ, vertical component u sin θ.
- Time to the maximum height: v = u + at with v = 0 for the vertical motion. On level ground the time of flight is twice this.
- Horizontal distance = horizontal component × time of flight.
- The speed at any point is found from the two components by Pythagoras.
Quantities and units
| Quantity | Symbol | Unit |
|---|---|---|
| Force | F | N |
| Resultant force | R | N |
| Tension | T | N |
| Angle | θ | ° |
| Initial velocity | u | m s−1 |
| Horizontal and vertical components of velocity | vH, vV | m s−1 |
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