IB DP Maths: AI HL复习笔记3.9.1 Kinematics with Vectors

How are vectors related to kinematics?

• Kinematics is the use of mathematics to model motion in objects
• If an object is moving in one dimension then its velocity, displacement and time are related using the formula s = vt
  • where s is displacement, v is velocity and t is the time taken
• If an object is moving in more than one dimension then vectors are needed to represent its velocity and displacement
  • Whilst time is a scalar quantity, displacement and velocity are both vector quantities
• Vectors are often used in questions in the context of forces, acceleration or velocity
• The position of an object at a particular time can be modelled using a vector equation

How do I find the direction of a vector?

• Vectors have opposite directions if they are the same size but opposite signs
• The direction of a vector is what makes it more than just a scalar
  • E.g. two objects with velocities of 7 m/s and ‑7 m/s are travelling at the same speed but in opposite directions
• Two vectors are parallel if and only if one is a scalar multiple of the other
• For real-life contexts such as mechanics, direction can be calculated from a given vector using trigonometry 
  • Given the i and j components a right-triangle can be created and the angle found using SOHCAHTOA
• It is usually given as a bearing or as an angle calculated anticlockwise from the positive x-axis

How do I find the distance between two moving objects?

• If two objects are moving with constant velocity in non-parallel directions the distance between them will change
• The distance between them can be found by finding the magnitude of their position vectors at any point in time
• The shortest distance between the two objects at a particular time can be found by finding the value of the time at which the magnitude is at its minimum value
  • Let the time when the objects are at the shortest distance be t
  • Find the distance, d, in terms of t by substituting into the equation for the magnitude of their position vectors
  • d² will be an expression in terms of t which can be differentiated and set to 0
  • Solving this will give the time at which the distance is at a minimum
  • Substitute this back into the expression for d to find the shortest distance