General Physics I Lecture 2: Motion in High Dimensions Prof. WAN, Xin 万歆 xinwan@zju.edu.cn http://zimp.zju.edu.cn/~xinwan/
Motion in 2D at Discrete Time
Do It the High School Way Along x direction, we have a constant velocity v0. x=x 0 +v 0 t Along -y direction, we have a constant acceleration g (assuming the initial velocity 0). v y = g t 2 y= y 0 g t /2 You have be careful with Which formula to use Sign of the velocity and acceleration
Choosing Different Axes y v x =v 0 cos θ g t sin θ 2 x=x 0 '+v 0 t cosθ g t sin θ/2 θ x v0 v y = v 0 sin θ g t cosθ g 2 y= y 0 ' v 0 t sin θ g t cos θ/2 We will come back to see how to represent the rotation. Need the concept of matrix.
Like more? Well, we can generalize the formulas to three dimensions. Sometimes (like in the theory of relativity), we need to write formulas for 3+1D, including the time axis. In thermodynamics, we will also use the space of the positions of many particles.
The More Educated Way We can unify the formulas we need for each direction to one single set. The cost is we have to learn some new math, namely, vector. = v0 + a t v r =r0 + v0 t + 1 2 t a 2 We do not need to know which direction the axes are, and we can forget about the signs and relax.
Vector A quantity that has both direction and magnitude and also obeys the laws of vector addition. boldface Examples: r, v, a, r,v a Physics laws with vectors are independent of the choice of the coordinate axes. No coordinate system is necessary. Vector notation is concise. Usually allows a simple and transparent formulation.
Vector versus Scalar A quantity having magnitude but not direction is a scalar. Examples: T (temperature), m (mass) The magnitude of a vector is a scalar. r = r displacement velocity speed acceleration distance
Magnitude and Unit Vector The magnitude of the vector A is written either A or A. A unit vector is a dimensionless vector having a magnitude of exactly 1. r r = r r Two vectors A and B may be defined to be equal if they have the same magnitude and point in the same direction. r r =r r
We Need Vector Algebra I have not define the necessary algebra. = v0 + a t v r =r0 + v0 t+ 1 2 t a 2 What do we need? Vector addition/subtraction Vector multiplied by a scalar
Addition R x = A x + Bx R y = A y + By R z = Az + B z parallelogram method triangular method
Addition Laws Associative law of addition A + (B + C) = (A + B) + C Commutative law of addition A+B=B+A
Finite Rotations Are Not Vectors The commutative law of addition is not satisfied by finite rotations. Not everything with a magnitude and a direction is a vector!
Negative and Subtraction The negative of the vector B is defined as the vector that when added to B gives zero for the vector sum. That is, B + (-B) = 0. The vectors B and B have the same magnitude but point in opposite directions. A - B = A + (-B)
Multiplication by a Scalar r R=3 r r r You can easily generalize to the case of a non-integer scalar. Q: What about the multiplication of two vectors?
Distributive Law Scalar multiplication is distributive over vector addition: B)=k k( A+ A+k B
Coordinate Systems x=r cos θ y=r sin θ
Back to Physics [Already learned] The motion of a particle moving along a straight line is completely known if its position is known as a function of time. [To be extended] The motion of a particle (moving in high dimensions) is completely known if its position vector r is known as a function of time.
Instantaneous Velocity The instantaneous velocity v is defined as the limit of the average velocity r/ t as t approaches zero: Note: dr and r are not in the same direction!
Back to Motion in 1D Do you find the difference?
Acceleration Vector
Constant Acceleration
Projectile Motion If you cannot figure out how to calculate the five curves, you will have to let me know or come at the office hour.
Uniform Circular Motion f vi v Δv = a = t f t i Δt Be careful with its direction. In the limit of t 0, we find the centripedal acceleration 2 v = r a r
Dawn of Physics Nicolaus Copernicus 1473-1543 Johanns Kepler 1571-1630 Galileo Galilei 1564-1642
Galileo's Workroom
Galileo's Ramp
Galileo's Discoveries Δ x t 2