Motion in Three Dimensions We ve learned about the relationship between position, velocity and acceleration in one dimension Now we need to extend those ideas to the three-dimensional world In the 1-D case, we described x, v, and a with simple numbers Sign indicated direction In three dimensions, there are an infinite number of possible directions, so We represent position, velocity, and acceleration with vectors A vector can be represented by an arrow: Tail Length indicates magnitude Head
ector Basics The arrow picture of a vector embodies the concepts we need Size and direction But we still need to treat these object mathematically Symbol for a vector is or Length is written as Can start by looking at components of the vector 2-D case: x y θ ( x, y ) Can choose x and y axes at our convenience y x x = cosθ y = sinθ
3-D case (see diagram on next page): 1. Choose one of the axes, and find the angle between the vector and that axis We typically choose the z axis, and call the angle θ 2. The vector and the axis chosen above form a plane. Find the angle between that plane and one of the other two axes We typically choose the x axis, and call this angle φ 3. The components of a 3-D vector are: x = sinθ cosϕ y = sinθ sinϕ z = cosθ
z xz θ y x φ y x
ector Component Relations The magnitude of the vector is given by the sum, in quadrature, of the components: = v + v + v 2 2 2 x y z The direction is also determined uniquely by the components: φ v 1 y = tan v x v + v θ = tan ; 0 < θ < 180 vz 2 2 1 x y o Take note of signs of v x and v y to get the correct quadrant
ector Addition We say that is the sum of its components: = x + y For this to work, we need to define what + means for vectors Graphical representation: B A + = B A + B A Reasonable? A + B = B + A A + (-A) = A A = 0 Intuitive? If one walks along A, and then along B, one arrives at the position given by A+B
One can also add vectors mathematically, using components: Let A = (A x, A y, A z ) and B = (B x, B y, B z ) Then A + B is: (A x + B x )i + (A y + B y )j + (A z + B z )k i,j, and k are vectors of length 1 along the x, y, and z directions -- we call them unit vectors Multiplication: Muliplying a vector by a number is straightforward: ca = ca x i + ca y j + ca z k Simply makes vector longer or shorter (and reverses direction if c < 0) What about multiplying two vectors? We ll get to that later
Kinematics with ectors We now consider a particle moving in three dimensions It s location at any time is given by a position vector r: z r(t ) r r(t) x y The average velocity between times t and t is: v avg ( t ) ( t) r r r = = t t t
From the definition of vector addition: So, ( ( ) ( )) ( ) ( ) ( ) ( ) ( ) ( ) r = r t r t i + r t r t j + r t r t k x x y y z z = r i + r j + r k x y z r r r r t t t t x y z = i + j+ k But we usually want to know the instantaneous velocity: dr dr dr x y drz v( t) = = i + j + k dt dt dt dt
Similarly, acceleration is given by: dv dv dv x y dvz a( t) = = i + j + k dt dt dt dt 2 2 2 d r d r x y d rz = i + j+ k 2 2 2 dt dt dt In summary, each component of the three-dimensional kinematic vectors follows the rules for one-dimensional motion Can think of it as three sets of 1-dimensional equations
Example A particle starts from rest at r = 0, and undergoes an acceleration given by a = Ati + Bcos(Ct)j. Where is the particle at time t? We need to know r(t), and the first step is finding v(t): v ( ) = a( ) t t dt ( ) = Ati + Bsin Ct j dt 1 2 B = At + c1i + sin( Ct) + c2j 2 C From the initial conditions, we know that c 1 = c 2 = 0
One more integration gives us: r ( ) = v( ) t t dt 1 B At i sin( Ct) j dt 2 C 2 = + 1 = At 6 3 B i C cos ( Ct) B C 2 2 j
With A = 1m/s 3, B = 100m/s 2 and C = 1/s, this motion looks like: 300 y(m) 200 100 0 0 50 100 150 200 x(m)
The Real Definition of a ector ectors are defined by their behavior under a rotation of the coordinate system Magnitude remains constant, but direction changes in response to rotation This implies a linear transformation of the vector components under rotation Any object whose components transform linearly when the coordinate system is changed is called a tensor ectors are a special case of tensors as are scalars, defined as quantities that don t change at all when the coordinate system changes Temperature, for example, is a scalar We ll only deal with scalars and vectors in this course