Overview
This lecture covers vectors and their properties, types of vector addition, projectile motion concepts and formulas, river crossing problems, and basics of circular motion, focusing on key definitions, formulas, and problem-solving approaches.
Vector Basics
- Vectors have both magnitude and direction; scalars have only magnitude.
- Position vectors describe location with respect to the origin and an angle θ.
- Vectors can be represented using x, y (and z) components.
- Equal vectors have the same magnitude and direction; parallel vectors have the same or opposite directions.
- Collinear vectors lie along the same line; coplanar vectors lie in the same plane.
- Concurrent vectors pass through a single point.
Vector Addition & Laws
- Triangular Law: Place vectors head-to-tail; the resultant vector joins the tail of the first to the head of the last.
- Parallelogram Law: Two vectors form adjacent sides of a parallelogram; the diagonal is the resultant.
- Resultant magnitude: ( R = \sqrt{A^2 + B^2 + 2AB\cos\theta} )
- Resultant direction: ( \tan\alpha = \frac{B\sin\theta}{A + B\cos\theta} )
Vector Product Operations
- Dot Product: ( \vec{A} \cdot \vec{B} = |\vec{A}||\vec{B}|\cos\theta ); gives a scalar.
- Cross Product: ( \vec{A} \times \vec{B} = |\vec{A}||\vec{B}|\sin\theta ); gives a vector using the right-hand rule.
- Dot product is zero if vectors are perpendicular; cross product is zero if vectors are parallel.
Resolution of Vectors
- Any vector can be split into horizontal (x) and vertical (y) components: ( x = r\cos\theta,, y = r\sin\theta ).
- Direction cosines relate vector components to their angles with coordinate axes.
Projectile Motion
- Path of a projectile is a parabola under gravity.
- Time of flight: ( T = \frac{2u\sin\theta}{g} )
- Maximum height: ( H = \frac{u^2\sin^2\theta}{2g} )
- Horizontal range: ( R = \frac{u^2\sin2\theta}{g} )
- Maximum range at θ = 45°; complementary angles yield same range.
- Trajectory equation: ( y = x\tan\theta - \frac{gx^2}{2u^2\cos^2\theta} )
- Velocity at any point: ( v = \sqrt{v_x^2 + v_y^2} )
River Crossing & Relative Motion
- Relative velocity concepts apply for moving boats or swimmers across rivers.
- Minimum time path is perpendicular to current; shortest path adjusts angle.
- Time to cross: ( t = \frac{\text{width}}{\sqrt{v_{\text{boat}}^2 - v_{\text{river}}^2}} )
Circular Motion
- Uniform circular motion: Constant speed, changing direction.
- Angular displacement θ in radians; angular velocity ω = θ/t.
- Centripetal acceleration: ( a_c = \frac{v^2}{r} ), acts toward center.
- Tangential acceleration changes speed along the circular path.
- Relationships: ( v = r\omega ), ( s = r\theta )
- Number of rotations: ( n = \frac{\theta}{2\pi} )
Key Terms & Definitions
- Vector — Quantity with magnitude and direction.
- Scalar — Quantity with magnitude only.
- Dot Product — Scalar product of two vectors.
- Cross Product — Vector product, perpendicular to both vectors.
- Centripetal Acceleration — Acceleration directed toward the center of a circle.
Action Items / Next Steps
- Review formulas for vector operations, projectile motion, and circular motion.
- Practice numerical problems covering projectile range, river crossing, and centripetal force.
- Prepare for next session on Newton’s Laws of Motion.