Motion in Two and three dimensions

• The position vector of an object with coordinates (x,y,z) can be written as: r=xi+yj+zk

• i,j,k are “unit vectors”; dimensionless vectors with a magnitude = one that point in the direction of the +x, +y, and +z axis respectively

• Example:

position vector: r=5i+10j+0k corresponds with the coordinates (5,10)

Position and Displacement

Position and Displacement continued

• Displacement

• Δr =r2-r1• In unit vector notation

Δr=(x2+x1)i+(y2+y1)j+(z2+z1)k

or Δr=xi+yj+zk ˆ

→ → →

This slide needs to be fixed!!

Average and Instantaneous Velocity

• Average velocity= Δr/Δt=displacement

Interval of time

In vector components

Vavg.= (Δx/Δt)i+ (Δy/Δt)j+(Δz/ Δt)k

Instantaneous velocity-as time interval shrinks to zero

V=dr/dt

→

→

Velocity continued

• The direction of the instantaneous velocity of a particle is always tangent to the particle’s path at its position.

• In unit vector notation:

v=(dx/dt)i+(dy/dt)j+(dz/dt)k→

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Average Acceleration and Instantaneous Acceleration

• Average acceleration=change in velocity

change in time

As Δt approaches 0 you get instantaneous acceleration: a=dv/dt=d2r/dt2

In unit vector notation:

a=axi+ayj+azk

Projectile Motion

• Projectile- a particle that moves in a vertical plane with some initial velocity but its acceleration is always the free-fall acceleration (downward)

• Initial velocity=voxi+voyj• Components voxand voy can be found if the

angle between the initial velocity and the positive x-axis

• vox=vocosθ and voy =vosinθ

Projectile Motion Continued

• In projectile motion the horizontal and vertical motion are independent of each other.

• Projectile motion problems can be solved by dividing the problem into horizontal motion (acceleration=zero, velocity is constant) and vertical motion (constant downward acceleration)

Projectile Motion Continued

Horizontal Motion:

• Vx remains unchanged from vi

dh=x-xi=vixt (vix=vicosθ)

dh=(vicosθ)

Vertical Motion:

dv =y-yi=viyt-(1/2)gt2=(visinθ)t-(1/2)gt2

Can you find the small mistake on this slide?

Uniform Circular Motion

• A particle is in uniform circular motion if it travels around a circle or a circular arc at uniform speed

• Although there is no change in speed the particle is constantly accelerating because there is constantly change in direction

• Centripetal motion-acceleration is always directed radially inward

• a=v2/r T=2πr/v

Relative Motion in One and Two Dimensions

• The velocity of a particle depends on the reference frame

• Usually velocity is measured relative to the ground

• xbc=xba+xac (two outside subscripts)

• rbc=rba+rac

• vbc=vba+vac

Comprehension Questions

• What is the horizontal velocity at the top of the trajectory?

Vx

Make this problem more specific!

• What is the vertical component of velocity at the top of the trajectory?

0 m/s

Comprehension Questions

• In projectile motion are the horizontal and vertical components dependant on each other? Why or why not?

No, because there is an acceleration in only one of the components and gravity affects only one of the components.

• Gravity affects which velocity component of projectile motion?

vertical component

• Which velocity component has constant acceleration and in what direction is the magnitude?

vertical component due to gravity in the downward direction

• What is the horizontal acceleration component of an object in projectile motion?

0 m/s^2

• Does an object in uniform circular motion have an acceleration?

yes

• Is that acceleration caused by a change in speed? If not what is it caused by?

no, there is a change in direction causing a change in velocity which means there is an acceleration

• What is the formula for centripetal acceleration?

a = v2/a

Fix the mistake!

• What is the formula for centripetal force?

Fc = mv2/r

• What is the formula for the period of revolution?

T=2πr/v

• In what direction does the centripetal acceleration point?

towards the center

• What direction is the centripetal force pointing towards?

towards the center

• In what direction relative to the centripetal acceleration does the velocity always point?

perpendicular to the accceleration

• In what direction relative to the circle does the velocity always point?

tangent to the circle

• What are the vector components of the velocity?

v = vxi + vyj

• What are the vector components of centripetal acceleration?

a = axi + ayj

• What is the most common reference frame?

ground

• What is the equation for relative motion?

XP-A = XP-B + XB-A

• What is the equation for average velocity and instantaneous velocity?

vavg = change in r / change in t

v = dr/dt

• What are the formulas for average acceleration and instantaneous acceleration?

aavg = change in v / change in t

a = dv/dt

Sample questions

• The original position of a particle is r=(-3m)i +(2m)j+(5m)k, and then the position is r2=(9m)i+(2m)j+(8m)k?

What is the question??

• A particle with velocity v0=-2i+4j ( in meters per second) at t=0 undergoes a constant acceleration of magnitude 30m/s2

at an angle of 130 degrees from the positive x-axis. What is the particle’s velocity at t=5 s in unit vector notation?

Good question—also find the displacement between 3 and 5 seconds.

• A particle moves so that its position ( in meters) as a function of time is r= i+4t2j+tk. Write expressions for its acceleration and velocity as functions of time.

Got an answer for this one?

• A car moving at a constant 60km/h moves east for 40 min. then in a direction 50 degrees east of due north for 20 min. and then west for 50min. What are the magnitude and angle of its average velocity?

• A small ball rolls horizontally off the edge of a table that is 1.20 m high. It hits the floor at 1.52 m from the edge. How long was it in the air?

Also find the speed and angle at which it hits!!

• A rotating fan completes 1200 rev. per minute. The radius of the blade is 0.15m. What distance does the tip move in one rev.?

• A fly ball has v= 25 i+ 4.9j ( in meters per second). Has the ball passed its highest point?

• A particle has a displacement of Δr= 2.0 i-3.0 j +6.0 k and has a final position of r=3.0j-4.0k. What was its initial position vector?

• A ball has an initial speed of 25m/s.what are the least and greatest elevation angles at which the ball can be kicked to score a field goal from a point 50 min front of the goal post whose horizontal bar is 3.44 m above the ground.

A projectile’s launch speed is 5X the speed at its max height. Find the launch angle.

Ok, these are good questions theo…how about solutions so people can check their answers?