Mechanics
Mechanics Motion along a Straight Line Displacement, Velocity, Acceleration • Displacement is the net distance with direction, of object from origin • Velocity is rate of change of displacement over time • Acceleration is rate of change of velocity over time Motion with constant Acceleration • Acceleration cannot be instantaneous as infinite acceleration contradicts Newton’s Second Law • All motion with constant acceleration, uses definitions of acceleration as 𝑎⃗ = •
𝑑𝑣⃗ 𝑑𝑣⃗ = 𝑣⃗ 𝑑𝑡 𝑑𝑥⃗
Displacement in axis are given by 𝑥⃗ = 𝑥⃗0 + 𝑣⃗𝑥0 𝑡 +
⃗⃗⃗⃗⃗𝑥 2 𝑎 𝑡 2
𝑦⃗ = 𝑦⃗0 + 𝑣⃗𝑦0 𝑡 +
⃗⃗⃗⃗⃗ 𝑎 𝑦 2 𝑡 2
Vectors Vectors, Resolution Vectors, Unit vectors • Vectors reflect direction and magnitude of properties of particles, e.g. force, acceleration • Resolution vectors are the components of a vector on different planes that sum to form it • Unit vectors are vectors on each plane with unit length o 𝑖̂ – unit vector in x direction o 𝑗̂ – unit vector in y direction o 𝑘̂ – unit vector in z direction Vector Addition, Dot/Scalar Product • Vectors are added tip to tail • Dot/ Scalar Product given by 𝑎⃗ . 𝑏⃗⃗ = |𝑎|. |𝑏| cos(∅) = ∑ 𝑎𝑖 𝑏𝑖
3D and 2D Motion Vector Form Equations 𝑎⃗ ⃗⃗⃗⃗𝑖̂ = 𝑥⃗0 𝑖̂ + 𝑣⃗0 𝑡𝑖̂ + 𝑖̂ 𝑡 2 𝑣 2 𝑎⃗ 𝑣⃗𝑘̂ = 𝑥⃗0 𝑘̂ + 𝑣⃗0 𝑡𝑘̂ + 𝑘̂ 𝑡 2 2
𝑑𝑣⃗ 𝑑𝑣⃗ = 𝑣⃗ 𝑑𝑡 𝑑𝑥⃗
Displacement in axis are given by 𝑥⃗ = 𝑥⃗0 + 𝑣⃗𝑥0 𝑡 +
⃗⃗⃗⃗⃗𝑥 2 𝑎 𝑡 2
𝑦⃗ = 𝑦⃗0 + 𝑣⃗𝑦0 𝑡 +
⃗⃗⃗⃗⃗ 𝑎 𝑦 2 𝑡 2
Vectors Vectors, Resolution Vectors, Unit vectors • Vectors reflect direction and magnitude of properties of particles, e.g. force, acceleration • Resolution vectors are the components of a vector on different planes that sum to form it • Unit vectors are vectors on each plane with unit length o 𝑖̂ – unit vector in x direction o 𝑗̂ – unit vector in y direction o 𝑘̂ – unit vector in z direction Vector Addition, Dot/Scalar Product • Vectors are added tip to tail • Dot/ Scalar Product given by 𝑎⃗ . 𝑏⃗⃗ = |𝑎|. |𝑏| cos(∅) = ∑ 𝑎𝑖 𝑏𝑖
3D and 2D Motion Vector Form Equations 𝑎⃗ ⃗⃗⃗⃗𝑖̂ = 𝑥⃗0 𝑖̂ + 𝑣⃗0 𝑡𝑖̂ + 𝑖̂ 𝑡 2 𝑣 2 𝑎⃗ 𝑣⃗𝑘̂ = 𝑥⃗0 𝑘̂ + 𝑣⃗0 𝑡𝑘̂ + 𝑘̂ 𝑡 2 2
