Resultant of Two Vectors

Physical Quantities and Measurement Techniques

Resultant of Two Vectors

• The resultant velocity of an airplane flying at 80 km/h in a 60 km/h crosswind can be found graphically or by calculation.
• Graphically: Set a scale (e.g., 1 cm = 10 km/h) and draw vectors accordingly.
• Calculation: If the vectors are at right angles, use the Pythagorean theorem: v² = 80² + 60²; v = 100 km/h.
• The direction of the resultant velocity can be found using trigonometry: tan⁻¹(60/80) ≈ 53°.

Resultant Vector

• The resultant vector is the total or net vector when multiple vectors are combined.
• The magnitude indicates size or amount, and direction indicates the way vectors point.

Adding Vectors

• Triangle Method: Link the head of vector a to the tail of vector b.
• The resultant vector connects the tail of a to the head of b.
• Parallelogram Method: Link the tails of vectors a and b, drawing a parallelogram with them as sides.
• The resultant vector is the diagonal of this parallelogram.

Motion - Acceleration

• Acceleration is the change in velocity per unit of time and is a vector with both magnitude and direction.
• Acceleration Formula: a = (v - u) / t, where a is acceleration (m/s²), v is final velocity (m/s), u is initial velocity (m/s), and t is time (s).
• Positive acceleration occurs when velocity increases (v > u).
• Negative acceleration (Deceleration) occurs when velocity decreases (v < u).
• No acceleration (Zero Acceleration) occurs when a car maintains a constant speed (u = v).

Motion - Speed and Velocity

• Speed is the distance moved per unit of time, and a scalar quantity, with magnitude only.
• Speed is measured in meters per second (m/s).
• Velocity is the change in displacement per unit of time, and a vector, with both magnitude and direction.
• Velocity is measured in meters per second (m/s).
• The sign indicates the direction of the object's motion.
• When an object travels in a straight line without changing direction, the magnitude of velocity equals speed.
• Formula for average speed: average speed = (u + v)/2
• Special case of average speed (u = v): (v + v)/2 = v

Motion - Distance and Displacement

• Distance is the total length of the path traveled by an object, a scalar quantity measured in meters (m).
• For an object moving from point A to B, distance is the half of the circumference of circle.
• Displacement is the directed length of the shortest path from the start to the end points, a vector quantity measured in meters (m).
• For an object moving from point A to B, displacement is the direct distance (diameter) from A to B with a specified direction.
• For an object moving from point A to B to A, displacement is zero.

Measurement Techniques

• Zero Error: Caused by faulty equipment that doesn't reset to zero properly. To minimize it, calibrate your instrument before use.
• Parallax Error: Occurs due to incorrect position of eyes. The objects shouldn't be touching the measurement scale. View object at perpendicular angle to the scale for avoiding parallax error.

Techniques - Measuring Small Lengths

• Ball Bearing Diameter: Place ball bearings between blocks, measure total diameter, then divide by amount of bearings.
• Paper Thickness (Ruler): Measure thickness of multiple sheets, then divide by the number of sheets.
• Paper Thickness (Micrometer): Check for zero error. Measure the thickness of several sheets of paper, and divide thickness by sheet amount.

Measuring Volume

• For measuring volume with measuring cylinder, read the value (V₂, V₁) at the bottom of the meniscus.
• Measurement should be done from eye-level.
• Volume of object can be calculate as: V=V₂-V₁

Period of Pendulum Measurement

• Measure the Time Period for amount of oscillations and repeat the reading to improve accuracy and reduce inaccuracy.
• Ensure to set stopwatch at zero to avoid any zero error.
• Divide total time by amount of swings to get time elapsed accurately.