Motion in 2D Vectors Projectile Motion

Motion in 2D

o Vectorso Projectile Motion

Physics -101Piri Reis University 2010

o Aim of the lecture Cover material relevant to 2-D and 3-D kinematics:

speed, acceleration vector representation projectile motion

o  Main learning outcomes familiarity with

vectors vector algebra and calculus equations of motion using vectors projectile motion

ability to solve kinetics problems in multiple dimensions compute projectile trajectories

Lecture 3

Review of some Mathematics

Vectors written as a a a a a & othersThey all mean the same thing – a direction and magnitude

X (east)

y (north)

The vector (5,4) means

5 10

5

10

X (east)

y (north)

The vector (5,4) means

5 10

5

10

5 km in the x direction (east)

X (east)

y (north)

The vector (5,4) means

5 10

5

10

4 km in the y direction (north)(5,4) to

tal

X (east)

y (north)

5 10

5

10Can also make (5,4) by going north first:

(5,4) is the sum of (5,0) and (0,4)And the order does not matter

(5,4) = (5,0) + (0,4) = (0,4) + (5,0)

X (east)

y (north)

In fact (5,4) can be made by adding lots ofvectors:

5 10

5

10

(5,4)

The sum of lots of ‘short walks’is the shortest distance betweenThe starting and ending place.

X (east)

y (north)

What does it mean to subtract vectors?

5 10

5

10

a

b-b

a+ba-b

Subtracting means togo in the opposite directionbut for the same distance.

Two vectors are added by adding the components

a = (3,4) b = (6,9) then a + b = (3+6,4+9)

Similarly to subtract them a – b = (3-6,4-9)

X (east)

y (north)

5 10

5

10

a

b

a+b

a-b

Review of some Mathematics

Review of some Mathematics

Review of some Mathematics

Its usual to usea ˆ above unitvectors, but not always, eg i,j

Review of some Mathematics

Vectors in 3-D

Vectors in 3-D are an extension of vectors in 2-D

Three components r = (x,y,z)

o The formalism is identical There is no way to tell from a vector equation if it is 2-D or 3-D It doesn’t really matter because all the algebra is the same The same relationships exist between vectors The same rules, but with one more component This is one of the advantages of working with vectors

• effectively you are manipulating all the dimensions simultaneously

z

x

y

z

x

yr

Review of some Mathematics

Product of vectors

Multiplying vectors is more complicated than multiplying simple numbers

o Multiplying by a scalar simply keeps the direction the same, but makes it longer:

if a = (3,5) then a = (3,5)

o The scalar product, written a.b can be defined as

a.b = IaI IbI cos() where is the angle between the two vector directions

o The scalar product is also called the ‘dot’ producto The scalar product is a scalar, it is an ordinary numbero The scalar product can also be evaluated from components:

If a = (7,2) and b = (4,9), then a.b = 7x4 + 2x9 Generally a.b = a1b1 + a2b2 where a = (a1,a2) and b = (b1,b2)

Motion in 2-D - Projectiles

X (east)

y (north)

5 10

5

10

r1 r2

r = r2-r1

If a particle starts from position r1 at time t = t1

and moves to position r2 at time t = t2

then its displacement vector is

r = r2-r1

which it moved in time t = t2 – t1

Motion in 2-D - Projectiles

X (east)

y (north)

5 10

5

10

r1 r2

r = r2-r1

Its average velocity is then defined as v = r / t

Velocity is a vector quantity, • its magnitude IvI is the speed of motion• its direction is the direction of motion.

Motion in 2-D - Projectiles

X (east)

y (north)

5 10

5

10

r1

rd

r = r2-r1

As the change becomes small, part of a path then v = r / t

becomes the instantaneous velocity

v = dr/dt

which is the derivative of displacement withrespect to time.

r = rd-r1

Motion in 2-D - Projectiles

X (east)

y (north)

5 10

5

10 The tangent to the curveis the direction of the velocity the magnitude CANNOT be determined from this plot as the time is not given

Consider a particle followingthe dotted trajectory

Here the particle stopped going south andstarted to go north

Projectiles

Motion

if an object is in motion, its position can be described by the equation

x = vt + x0

where x is its location at time t, v is its velocity, x0 its starting displacement

if the velocity is not constant then

x = x0 + ∫ vdt (this was discussed last week)

A special case is when the acceleration, a, is constant, in which case

x = x0 + ∫ vdt = x0 + v0t + ½at2 where v0 is the starting velocity

or x = v0t + ½at2

this is often written as s = ut + ½at2 for the distance traveled in time t in 1-D

Projectiles

Motion

These equations all have 1-D equivalents – vectors are just a ‘shorthand’

x = vt + x0 is the same as x = vxt = x0 and y = vyt + y0

where x, y are locations at time t, x0 starting x position etc..

if the velocity is not constant then

x = x0 + ∫ vxdt and y = y0 + ∫ vydt

A special case is when the acceleration is constant, in which case

x = x0 + ∫ vxdt = x0 + vx0t + ½axt2 & y = y0 + ∫ vydt = y0 + vy0t + ½ayt2

or x = vx0t + ½axt2 & y = vy0t + ½ayt2

Each dimension has its own set of equations BUT TIME IS SAME FOR BOTH

This is NOT whatfalling apples do,

They fall directlydownwards andthe reason isgravity.

They do howeverget faster and fasteras they fall, they move with constantacceleration

Falling apples obey the equations

y = y0 + ½at2

v = at

but a is caused by gravityand is -9.81 m/s2

we use the symbol gfor the acceleration causedby gravity at the surface of the earth.

The minus sign just meansthat the acceleration isdownwards

y = y0 + ½at2

v = gt2

so y = 8 – 9.81t2 m 2 When the apple hits the ground

0 = 8 – 9.81t2

2 so t = √(16 / 9.81) = 1.28 s

8m

y = y0 + ½at2

v = gt

more generally

t = √(2y0/g)

so the speed of impact on the ground is vground = gt = g√(2y0/g) = √(2gy0)

Check the dimensions v - in [m/s]√(2gy0) g in [ms-2] y0 in [m] so √([ms-2][m])

2-D projectile

In the x direction the equation of motion is

x = vx0t vx = vx0 (constant speed in x direction)

In the y direction it is the same as for the falling apple, except thatthe initial velocity is not zero and the apple starts from ground level

For falling apple: y = y0 + ½at2

vy = atBut now y0 is 0 as the apple starts from groundAnd the initial velocity, vy0 is no longer zero. So:

y = vy0t + ½at2

vy = vy0 + at

2-D projectile

y = vy0t + ½at2

vy = vy0 + at

x = vx0t vx = vx0

t = x/vx0

y = vy0{x/vx0} + ½a{x/vx0}2

but a = g = -9.81 m/sso:

y = {vy0/vx0}x - {4.9/vx02}x2

Which describes a parabola

2-D projectile

y = vy0t + ½at2

vy = vy0 + at

x = vx0t vx = vx0

y = {vy0/vx0}x - {4.9/vx02}x2

If ax2+bx + c = 0 then x = -b±√(b2-4ac) 2aSo the value for x for a given y canbe evaluated by regarding y as aconstant and solving as a quadratic

vy = vy0 + gt = vy0 + g{x/vx0} = vy0 + {g/vx0}x = vy0 – {9.81/vx0}x

2-D projectile

y = {vy0/vx0}x - {4.9/vx02}x2

vy = vy0 - {9.81/vx0}x

vx = vx0Note that because of thenegative sign (which came from g)the vertical speed starts out +vebut becomes negative

The horizontal speed neverchanges – this assumes noair resistance

Projectile motion can be described using a vector equation instead

r = r0 + v0t + ½at2

x

y

r

More on gravity in anotherlecture

x

y

r

o There are several closely related projectile problems the differences are essentially the ‘boundary conditions’, which means the starting values. the equations are the same, but the values at the start are different

x

y

x

y