Projectile Motion September 12, 2012 - p. 1/10

September 12, Week 4

Physics 151

Today: Chapter 3: 2D Motion

Homework Assignment #4 - Due September 14.

Mastering Physics: 7 problems from chapter 3.

Written Questions: 3.4, 3.69

The motion diagram for problem 3.4 can be found in thehomework file on the webpage for convenient printing.

Thursday office hours, 2:00-6:00.

Exam #1 - Monday, September 17.

Practice Exam Available on Website.

Projectile Motion September 12, 2012 - p. 2/10

Two-Dimensional Motion

To describe motion, we still need to know position, velocity, andacceleration at all times.

Projectile Motion September 12, 2012 - p. 2/10

Two-Dimensional Motion

To describe motion, we still need to know position, velocity, andacceleration at all times.

In 2D, this means we have to know the components of theposition, velocity, and acceleration vectors.

Projectile Motion September 12, 2012 - p. 2/10

Two-Dimensional Motion

To describe motion, we still need to know position, velocity, andacceleration at all times.

In 2D, this means we have to know the components of theposition, velocity, and acceleration vectors.

To locate an object, we have to give two numbers: (x, y). Theyare the cartesian coordinates AND they are the components ofthe position vector.

Projectile Motion September 12, 2012 - p. 2/10

Two-Dimensional Motion

To describe motion, we still need to know position, velocity, andacceleration at all times.

In 2D, this means we have to know the components of theposition, velocity, and acceleration vectors.

To locate an object, we have to give two numbers: (x, y). Theyare the cartesian coordinates AND they are the components ofthe position vector.

b (x, y)

Projectile Motion September 12, 2012 - p. 2/10

Two-Dimensional Motion

To describe motion, we still need to know position, velocity, andacceleration at all times.

In 2D, this means we have to know the components of theposition, velocity, and acceleration vectors.

To locate an object, we have to give two numbers: (x, y). Theyare the cartesian coordinates AND they are the components ofthe position vector.

b (x, y)

−→r

Projectile Motion September 12, 2012 - p. 2/10

Two-Dimensional Motion

To describe motion, we still need to know position, velocity, andacceleration at all times.

In 2D, this means we have to know the components of theposition, velocity, and acceleration vectors.

To locate an object, we have to give two numbers: (x, y). Theyare the cartesian coordinates AND they are the components ofthe position vector.

b (x, y)

−→r

−→x

−→y

Projectile Motion September 12, 2012 - p. 3/10

Velocity Components

In curved motion, the path taken bya moving object is called its trajectory

x

y

b

b

b b

Projectile Motion September 12, 2012 - p. 3/10

Velocity Components

In curved motion, the path taken bya moving object is called its trajectory

x

y

b

b

b b

Projectile Motion September 12, 2012 - p. 3/10

Velocity Components

In curved motion, the path taken bya moving object is called its trajectory

x

y

b

b

b b

Projectile Motion September 12, 2012 - p. 3/10

Velocity Components

In curved motion, the path taken bya moving object is called its trajectory

x

y

b

b

b b

Projectile Motion September 12, 2012 - p. 3/10

Velocity Components

In curved motion, the path taken bya moving object is called its trajectory

x

y

b

b

b b

Projectile Motion September 12, 2012 - p. 3/10

Velocity Components

In curved motion, the path taken bya moving object is called its trajectory

x

y

b

b

b b

Projectile Motion September 12, 2012 - p. 3/10

Velocity Components

In curved motion, the path taken bya moving object is called its trajectory

x

y

b

b

b b

Projectile Motion September 12, 2012 - p. 3/10

Velocity Components

In curved motion, the path taken bya moving object is called its trajectory

x

y

Trajectory

b

b b

Projectile Motion September 12, 2012 - p. 3/10

Velocity Components

In curved motion, the path taken bya moving object is called its trajectory

There are two separate positionplots which give the velocityvector’s components

x

y

Trajectory

b

b b

Projectile Motion September 12, 2012 - p. 3/10

Velocity Components

In curved motion, the path taken bya moving object is called its trajectory

There are two separate positionplots which give the velocityvector’s components

x

y

Trajectoryb

t

x

b t

yb

b

b b

Projectile Motion September 12, 2012 - p. 3/10

Velocity Components

In curved motion, the path taken bya moving object is called its trajectory

There are two separate positionplots which give the velocityvector’s components

x

y

Trajectoryb

t

x

t

y

b

b b

Projectile Motion September 12, 2012 - p. 3/10

Velocity Components

In curved motion, the path taken bya moving object is called its trajectory

There are two separate positionplots which give the velocityvector’s components

x

y

Trajectoryb

t

x

t

y

b

b b

Projectile Motion September 12, 2012 - p. 3/10

Velocity Components

In curved motion, the path taken bya moving object is called its trajectory

There are two separate positionplots which give the velocityvector’s components

x

y

Trajectoryb

t

x

t

y

b

b b

Projectile Motion September 12, 2012 - p. 3/10

Velocity Components

In curved motion, the path taken bya moving object is called its trajectory

There are two separate positionplots which give the velocityvector’s components

x

y

Trajectory

b

t

x

t

y

b

b b

Projectile Motion September 12, 2012 - p. 3/10

Velocity Components

In curved motion, the path taken bya moving object is called its trajectory

There are two separate positionplots which give the velocityvector’s components

x

y

Trajectory

b

t

x

t

y

b

b b

Projectile Motion September 12, 2012 - p. 3/10

Velocity Components

In curved motion, the path taken bya moving object is called its trajectory

There are two separate positionplots which give the velocityvector’s components

x

y

Trajectory

b

t

x

t

y

b

b b

Projectile Motion September 12, 2012 - p. 3/10

Velocity Components

In curved motion, the path taken bya moving object is called its trajectory

There are two separate positionplots which give the velocityvector’s components

x

y

Trajectory

b

x

y

t

x

b Slope = vx

t

x

x-component

of velocityt

y

b Slope = vy

t

y

y-component

of velocity

Projectile Motion September 12, 2012 - p. 4/10

Curved Motion II

On the trajectory plot, the velocity vector is described as being“tangent" to the curve.

b

Projectile Motion September 12, 2012 - p. 4/10

Curved Motion II

On the trajectory plot, the velocity vector is described as being“tangent" to the curve.

The velocity vector is at the same angle as the slope of thetrajectory graph.

b

Projectile Motion September 12, 2012 - p. 4/10

Curved Motion II

On the trajectory plot, the velocity vector is described as being“tangent" to the curve.

The velocity vector is at the same angle as the slope of thetrajectory graph.

x

y

Trajectory

b

Projectile Motion September 12, 2012 - p. 4/10

Curved Motion II

On the trajectory plot, the velocity vector is described as being“tangent" to the curve.

The velocity vector is at the same angle as the slope of thetrajectory graph.

x

y

Trajectory

b

b

Projectile Motion September 12, 2012 - p. 4/10

Curved Motion II

On the trajectory plot, the velocity vector is described as being“tangent" to the curve.

The velocity vector is at the same angle as the slope of thetrajectory graph.

x

y

Trajectory

b

Line with sameslope as trajectory graph

b

Projectile Motion September 12, 2012 - p. 4/10

Curved Motion II

On the trajectory plot, the velocity vector is described as being“tangent" to the curve.

The velocity vector is at the same angle as the slope of thetrajectory graph.

x

y

Trajectory

b

Line with sameslope as trajectory graph

−→v

b

Projectile Motion September 12, 2012 - p. 4/10

Curved Motion II

On the trajectory plot, the velocity vector is described as being“tangent" to the curve.

The velocity vector is at the same angle as the slope of thetrajectory graph.

x

y

Trajectory

b

Line with sameslope as trajectory graph

−→v

vx

b

Projectile Motion September 12, 2012 - p. 4/10

Curved Motion II

On the trajectory plot, the velocity vector is described as being“tangent" to the curve.

The velocity vector is at the same angle as the slope of thetrajectory graph.

x

y

Trajectory

b

Line with sameslope as trajectory graph

−→v

vx

vy

b

Projectile Motion September 12, 2012 - p. 4/10

Curved Motion II

On the trajectory plot, the velocity vector is described as being“tangent" to the curve.

The velocity vector is at the same angle as the slope of thetrajectory graph.

x

y

Trajectory

b

Line with sameslope as trajectory graph

−→v

vx

vySpeed is the magnitudeof the velocity vector

⇒ v =√

v2x + v2y

Projectile Motion September 12, 2012 - p. 5/10

Acceleration Components

We can find the accelerationcomponents in the same wayas velocity

x

y

b

b

b

b

Projectile Motion September 12, 2012 - p. 5/10

Acceleration Components

We can find the accelerationcomponents in the same wayas velocity

x

y

b

b

b

b

Projectile Motion September 12, 2012 - p. 5/10

Acceleration Components

We can find the accelerationcomponents in the same wayas velocity

x

y

b

b

b

b

Projectile Motion September 12, 2012 - p. 5/10

Acceleration Components

We can find the accelerationcomponents in the same wayas velocity

x

y

b

b

b

b

Projectile Motion September 12, 2012 - p. 5/10

Acceleration Components

We can find the accelerationcomponents in the same wayas velocity

x

y

b

b

b

b

Projectile Motion September 12, 2012 - p. 5/10

Acceleration Components

We can find the accelerationcomponents in the same wayas velocity

x

y

b

b

b

b

Projectile Motion September 12, 2012 - p. 5/10

Acceleration Components

We can find the accelerationcomponents in the same wayas velocity

x

y

b

b

b

b

Projectile Motion September 12, 2012 - p. 5/10

Acceleration Components

We can find the accelerationcomponents in the same wayas velocity

x

y

Trajectory

b

b

b

Projectile Motion September 12, 2012 - p. 5/10

Acceleration Components

We can find the accelerationcomponents in the same wayas velocity

There are two separate VELOCITYplots which give the accelerationvector’s components

x

y

Trajectory

b

b

b

Projectile Motion September 12, 2012 - p. 5/10

Acceleration Components

We can find the accelerationcomponents in the same wayas velocity

There are two separate VELOCITYplots which give the accelerationvector’s components

x

y

Trajectoryb

t

vx

b

t

vy

b

b

b

b

Projectile Motion September 12, 2012 - p. 5/10

Acceleration Components

We can find the accelerationcomponents in the same wayas velocity

There are two separate VELOCITYplots which give the accelerationvector’s components

x

y

Trajectorybvx

vy

t

vx

t

vy

b

b

b

Projectile Motion September 12, 2012 - p. 5/10

Acceleration Components

We can find the accelerationcomponents in the same wayas velocity

There are two separate VELOCITYplots which give the accelerationvector’s components

x

y

Trajectorybvx

vy

t

vx

t

vy

b

b

b

Projectile Motion September 12, 2012 - p. 5/10

Acceleration Components

We can find the accelerationcomponents in the same wayas velocity

There are two separate VELOCITYplots which give the accelerationvector’s components

x

y

Trajectoryb

vxvy

t

vx

t

vy

b

b

b

Projectile Motion September 12, 2012 - p. 5/10

Acceleration Components

We can find the accelerationcomponents in the same wayas velocity

There are two separate VELOCITYplots which give the accelerationvector’s components

x

y

Trajectory

bvx

vy

t

vx

t

vy

b

b

b

Projectile Motion September 12, 2012 - p. 5/10

Acceleration Components

We can find the accelerationcomponents in the same wayas velocity

There are two separate VELOCITYplots which give the accelerationvector’s components

x

y

Trajectory

bvx

vy

t

vx

t

vy

b

b

b

Projectile Motion September 12, 2012 - p. 5/10

Acceleration Components

We can find the accelerationcomponents in the same wayas velocity

There are two separate VELOCITYplots which give the accelerationvector’s components

x

y

Trajectory

bvx

vy

t

vx

t

vy

b

b

b

Projectile Motion September 12, 2012 - p. 5/10

Acceleration Components

We can find the accelerationcomponents in the same wayas velocity

There are two separate VELOCITYplots which give the accelerationvector’s components

x

y

Trajectory

b

x

y

−→v

vx

vy

t

vx

b

Slope = ax

t

vx

x-component

of acceleration

t

vy

b

Slope = ay

t

vyy-component

of acceleration

Projectile Motion September 12, 2012 - p. 6/10

Projectile Motion

Projectile Motion is one example of two-dimensional motionwith a constant acceleration.

Projectile Motion September 12, 2012 - p. 6/10

Projectile Motion

Projectile Motion is one example of two-dimensional motionwith a constant acceleration.

Projectile - Any object that is launched into motion and thenacted on by gravity only.

Projectile Motion September 12, 2012 - p. 6/10

Projectile Motion

Projectile Motion is one example of two-dimensional motionwith a constant acceleration.

Projectile - Any object that is launched into motion and thenacted on by gravity only.

Ignore air resistance again.

Projectile Motion September 12, 2012 - p. 6/10

Projectile Motion

Projectile Motion is one example of two-dimensional motionwith a constant acceleration.

Projectile - Any object that is launched into motion and thenacted on by gravity only.

Ignore air resistance again.

Gravity pulls straight down, so it causes acceleration in they-direction only.

ax = 0, ay = −g (Down is negative )

Projectile Motion September 12, 2012 - p. 7/10

Projectile Equations

ax = 0 means that there is no change in the x-component ofvelocity ⇒ uniform motion in x.

Projectile Motion September 12, 2012 - p. 7/10

Projectile Equations

ax = 0 means that there is no change in the x-component ofvelocity ⇒ uniform motion in x.

ax = 0 =(vx)f − (vx)i

∆t

Projectile Motion September 12, 2012 - p. 7/10

Projectile Equations

ax = 0 means that there is no change in the x-component ofvelocity ⇒ uniform motion in x.

ax = 0 =(vx)f − (vx)i

∆t⇒ (vx)f = (vx)i ← no change

Projectile Motion September 12, 2012 - p. 7/10

Projectile Equations

ax = 0 means that there is no change in the x-component ofvelocity ⇒ uniform motion in x.

ax = 0 =(vx)f − (vx)i

∆t⇒ (vx)f = (vx)i ← no change

t

vx

Projectile Motion September 12, 2012 - p. 7/10

Projectile Equations

ax = 0 means that there is no change in the x-component ofvelocity ⇒ uniform motion in x.

ax = 0 =(vx)f − (vx)i

∆t⇒ (vx)f = (vx)i ← no change

t

vx

Projectile Motion September 12, 2012 - p. 7/10

Projectile Equations

ax = 0 means that there is no change in the x-component ofvelocity ⇒ uniform motion in x.

ax = 0 =(vx)f − (vx)i

∆t⇒ (vx)f = (vx)i ← no change

∆x = Area

t

vx

Projectile Motion September 12, 2012 - p. 7/10

Projectile Equations

ax = 0 means that there is no change in the x-component ofvelocity ⇒ uniform motion in x.

ax = 0 =(vx)f − (vx)i

∆t⇒ (vx)f = (vx)i ← no change

∆x = Area

t

vx

ti tf∆t

(vx)i

Projectile Motion September 12, 2012 - p. 7/10

Projectile Equations

ax = 0 means that there is no change in the x-component ofvelocity ⇒ uniform motion in x.

ax = 0 =(vx)f − (vx)i

∆t⇒ (vx)f = (vx)i ← no change

∆x = Area⇒ xf = xi + (vx)i∆t

t

vx

ti tf∆t

(vx)i

Projectile Motion September 12, 2012 - p. 8/10

Projectile Equations II

ay = −g means that there is change in the y-component ofvelocity. Gravity is constant ⇒ constant acceleration motion.

Projectile Motion September 12, 2012 - p. 8/10

Projectile Equations II

ay = −g means that there is change in the y-component ofvelocity. Gravity is constant ⇒ constant acceleration motion.

ay = −g =(vy)f − (vy)i

∆t

Projectile Motion September 12, 2012 - p. 8/10

Projectile Equations II

ay = −g means that there is change in the y-component ofvelocity. Gravity is constant ⇒ constant acceleration motion.

ay = −g =(vy)f − (vy)i

∆t⇒ (vy)f = (vy)i − g (∆t)

Projectile Motion September 12, 2012 - p. 8/10

Projectile Equations II

ay = −g means that there is change in the y-component ofvelocity. Gravity is constant ⇒ constant acceleration motion.

ay = −g =(vy)f − (vy)i

∆t⇒ (vy)f = (vy)i − g (∆t)

t

vy

Projectile Motion September 12, 2012 - p. 8/10

Projectile Equations II

ay = −g means that there is change in the y-component ofvelocity. Gravity is constant ⇒ constant acceleration motion.

ay = −g =(vy)f − (vy)i

∆t⇒ (vy)f = (vy)i − g (∆t)

t

vy

(vy)i

ti

(vy)f

tf

Projectile Motion September 12, 2012 - p. 8/10

Projectile Equations II

ay = −g means that there is change in the y-component ofvelocity. Gravity is constant ⇒ constant acceleration motion.

ay = −g =(vy)f − (vy)i

∆t⇒ (vy)f = (vy)i − g (∆t)

∆y = Area

t

vy

(vy)i

ti

(vy)f

tf

Projectile Motion September 12, 2012 - p. 8/10

Projectile Equations II

ay = −g means that there is change in the y-component ofvelocity. Gravity is constant ⇒ constant acceleration motion.

ay = −g =(vy)f − (vy)i

∆t⇒ (vy)f = (vy)i − g (∆t)

∆y = Area

t

vy

(vy)i

ti

(vy)f

tf

Projectile Motion September 12, 2012 - p. 8/10

Projectile Equations II

ay = −g means that there is change in the y-component ofvelocity. Gravity is constant ⇒ constant acceleration motion.

ay = −g =(vy)f − (vy)i

∆t⇒ (vy)f = (vy)i − g (∆t)

∆y = Area ⇒ yf = yi + (vy)i∆t−1

2(g)∆t2

t

vy

(vy)i

ti

(vy)f

tf

In both ofthese equationsg = +9.8m/s2.

Projectile Motion September 12, 2012 - p. 9/10

Launch Angle

(vx)i and (vy)i are the components of the initial velocity vector.Usually, we are given the launch speed, vi and angle, θ.

Projectile Motion September 12, 2012 - p. 9/10

Launch Angle

(vx)i and (vy)i are the components of the initial velocity vector.Usually, we are given the launch speed, vi and angle, θ.

−→vi

θ

vi = launch speed

θ = launch angle

Projectile Motion September 12, 2012 - p. 9/10

Launch Angle

(vx)i and (vy)i are the components of the initial velocity vector.Usually, we are given the launch speed, vi and angle, θ.

−→vi

θ

vi = launch speed

θ = launch angle

(vx)i = vi cos θ

Projectile Motion September 12, 2012 - p. 9/10

Launch Angle

(vx)i and (vy)i are the components of the initial velocity vector.Usually, we are given the launch speed, vi and angle, θ.

−→vi

θ

vi = launch speed

θ = launch angle

(vx)i = vi cos θ

(vy)i = vi sin θ

Projectile Motion September 12, 2012 - p. 10/10

Summary

Projectile Equations

ax = 0 ay = −g

(vx)f = (vx)i (vy)f = (vy)i − g∆t

xf = xi + (vx)i∆t yf = yi + (vy)i∆t− 1

2g∆t2

(vx)i = vi cos θ (vy)i = vi sin θ