Kinematics in 2-D. Review - What is Kinematics??? Describes the motion of objects Uses a set of equations Draws a relationship between time, distance,

Kinematics in 2-D

Review - What is Kinematics???

• Describes the motion of objects• Uses a set of equations• Draws a relationship between time, distance,

velocity, and acceleration

We Started in 1-D

• One dimensional motion (1-D)– Motion only in one plane (either in the x-direction

or the y-direction)– Motion in a straight line

Now Let’s Do it in 2-D

• Two Dimensional Motion– Instead of motion in either the x-direction or the

y-direction, we have motion in both the x-direction and the y-direction

– The motion is no longer in a straight line; it is parabolic

The Projectile

• Any object that travels through the air in which the only force acting on it is gravity

What are some example of projectiles?

And another…

Some Important Things to Know

• Projectiles travel in 2 dimensions, in the x-direction AND in the y-direction

• The ONLY force acting on a projectile is gravity• The motion in the x-direction is COMPLETELY

INDEPENDENT of the motion in the y-direction…let’s take a look

One More Look at Some Projectiles

Sign Conventions

• Positive directions (+d)– Right– Up

• Negative directions (-d)– Left– Down

• Now it is important as ever!!!

The Good News…

• The same equations from 1-D kinematics apply in 2-D

vtd

atvv if

2

2

1attvd i

advv if 222

Solving Problems

1. Draw a picture of the problem2. Define your sign convention

1. It is your decision, but use one that is convenient2. Typically, up is positive and right is positive

3. Separate the x-components and the y-components so you can solve them separately

4. Use the GUESS method

Remember g?

• Earlier, we said g for the surface of the Earth equals 9.81 m/s2

• We were not taking into consideration the direction in which g works

• What do you think g is?

2/81.9 smg

Crazy Important…

• The motion in the x-direction is COMPLETELY INDEPENDENT of the motion in the y-direction

• There is one link between the x-direction and the y-direction, and it is

t

Lingo and Such

Terms• Projectile• Trajectory• Range• Altitude• Hang time

Variables/Constants

• Vi,x or V0,x

• Vi,y or V0,y

• Vf,x

• Vf,y

• t• d or h• g

Example Problem

An airplane is flying with a constant velocity of 115 m/s. It drops a package from an altitude of 1050 m. If we ignore air resistance, how far down range will the object be when it hits the ground?

An airplane is flying with a constant velocity of 115 m/s. It drops a package from an altitude of 1050 m. If we ignore air resistance, how far down range will the object be when it hits the ground?

• Step 1: Draw a Picture

An airplane is flying with a constant velocity of 115 m/s. It drops a package from an altitude of 1050 m. If we ignore air resistance, how far down range will the object be when it hits the ground?

• Step 1: Draw a Picture• Step 2: Define a Sign

Convention

An airplane is flying with a constant velocity of 115 m/s. It drops a package from an altitude of 1050 m. If we ignore air resistance, how far down range will the object be when it hits the ground?

• Step 1: Draw a Picture• Step 2: Define a Sign

Convention• Step 3: Pick the

direction you want to start in and solve!

Let’s Start in the Y Direction

An airplane is flying with a constant velocity of 115 m/s. It drops a package from an altitude of 1050 m. If we ignore air resistance, how far down range will the object be when it hits the ground?

Given:d = -1050 mg = -9.8 m/s2

Unknown:t = ?

Equation:

2

2

1gtd

Let’s Start in the Y Direction

An airplane is flying with a constant velocity of 115 m/s. It drops a package from an altitude of 1050 m. If we ignore air resistance, how far down range will the object be when it hits the ground?

Substitute:

Solve:

2)8.9(2

11050 t

sec6.14t

2

2

1gtd

sec6.14t

Now Let’s Solve the X-Direction

An airplane is flying with a constant velocity of 115 m/s. It drops a package from an altitude of 1050 m. If we ignore air resistance, how far down range will the object be when it hits the ground?

Given:vi,x = +115 m/s

t = 14.6 secUnknown:

dx = ?

Equation:

vtd

Now Let’s Solve the X-Direction

An airplane is flying with a constant velocity of 115 m/s. It drops a package from an altitude of 1050 m. If we ignore air resistance, how far down range will the object be when it hits the ground?

Substitute:

Solve:

vtd

md 1679

)6.14)(115(d

1679 meters

Practice #1

A car drives off a cliff that is 50 m high. When it lands, investigators measure that the car is 85 m away from the base of the cliff. Calculate the following:

(1)the time the car was in the air.(2)the horizontal velocity of the car when it

drove off the cliff?

Practice #2

You throw a ball straight up with an initial vertical velocity of 22 m/s. Calculate the following:

(1)the time the ball is in the air.(2)how high the ball goes.