Two-dimensional motion

Curved motion

-or-

straight-line motion that is not aligned with a single axis.

+y (m)

+x (m) +1 +2 +3 +4 -4 -3 -2 -1

-1

-2

-3

-4

+4

+3

+2

+1

Position +y (m)

+x (m) +1 +2 +3 +4 -4 -3 -2 -1

-1

-2

-3

-4

+4

+3

+2

+1

Position is the (x,y) coordinates of an object on a 2-D Cartesian coordinate system.

x is the x-position.

y is the y-position.

Distance

Distance (from the origin) is found using the Pythagorean theorem.

+y (m)

+x (m) +1 +2 +3 +4 -4 -3 -2 -1

-1

-2

-3

-4

+4

+3

+2

+1

AngleMemorize the following trig functions.

+y (m)

+x (m) +1 +2 +3 +4 -4 -3 -2 -1

-1

-2

-3

-4

+4

+3

+2

+1

Angle (Example) +y (m)

+x (m) +1 +2 +3 +4 -4 -3 -2 -1

-1

-2

-3

-4

+4

+3

+2

+1

Converting distance and angle to position (coordinates)

A child pulls a sled with a 2-m long rope that makes an angle of 30° with respect to the ground. If the origin is at the point where the rope is tied to the sled, what is the position of the child’s hand?

-Sketch a picture.

-Draw the right triangle showing the x and y components and the given angle.

-Use trig functions to calculate the x and y components (i.e. legs of the right triangle).

Poll

The position of an object is at x = 3 m and y = 4 m. Which arrow most closely points in the direction of the object’s position?

Poll

Your table in the physics lab is 8.5 m from a certain corner of the room at an angle of 210 from East. What is the position of your table? (Use N as the +y axis and E as the +x axis).

1. x = 7.36 m, y = 4.25 m

2. x = +7.36 m, y = 4.25 m

3. x = 4.25 m, y = 7.36 m

4. x = 4.25 m, y = 7.36 m

5. x = 7.36 m, y = 4.25 m

6. x = 7.36 m, y = 4.25 m

Poll

The center of the bar, relative to a pole vaulter getting ready to run, is at the position x = 15 m, y=5 m. At what distance is the bar, relative to the pole vaulter?

1. 15 m

2. 20 m

3. 15.8 m

4. 250 m

5. 10 m

Displacement +y (m)

+x (m) +1 +2 +3 +4 -4 -3 -2 -1

-1

-2

-3

-4

+4

+3

+2

+1

Displacement is the vector (i.e. arrow) from an initial position to a final position.

Displacement tells you direction and distance an object moved.

Displacement is the “change in position” of an object.

Example

+y (m)

+x (m) +1 +2 +3 +4 -4 -3 -2 -1

-1

-2

-3

-4

+4

+3

+2

+1

What is the displacement of the toy car? Sketch the vector and write the x-displacement and y-displacement. What distance does it travel and at what angle?

PollA Volkswagon moves from position 1 to position 2. Which arrow most closely points in the direction of the displacement of the car?

+y (m)

+x (m) +1 +2 +3 +4 -4 -3 -2 -1

-1

-2

-3

-4

+4

+3

+2

+1

t1 t2

Poll

A soccer ball is displaced x = 14 m, y = 4 m where +x is “to the right” and +y is “forward toward the goal.” The displacement vector tells us that the soccer ball traveled…

1. to the right and toward the goal

2. to the left and toward the goal

3. to the left and away from the goal

4. to the left and toward the goal

5. straight to the right

6. straight to the left

7. straight toward the goal

8. straight away from the goal

Total Displacement

A soccer ball is kicked with a displacement x = 14 m, y = 4 m along the ground. It’s then kicked again with the displacement x = 4 m, y = 8 m. What is the total displacement of the ball from its original position?

To add vectors algebraically, add their components separately.

The resulting vector is called the resultant.

To add them pictorially, draw them tail to head, one after the other. The resultant is from the tail of the first vector to the head of the last vector.

Example

Chapter 1Vectors

Matter & InteractionsPHY 221 – Dr. Aaron Titus

A car travels from x = 4, y = 4 to x = 4, y=4 and then to x = 3, y = 3. How far is the car from where it started?

Poll

A car travels from position 1 to position 2 to position 3. Which is larger, the distance traveled or the magnitude of the total displacement?

1. distance traveled

2. magnitude of total displacement

3. neither, because they are equal

Average Velocity

In the same direction as displacement. Time interval may be large.

Intantaneous Velocity

In the same direction as displacement during a small time interval.

Instantaneous velocity is ALWAYS tangent to the path and in the direction of motion.

Instantaneous VelocityA pendulum swings back and forth. What is the pendulum’s direction of motion at the lowest point of its swing, as it travels from right to left?

Inst. velocity is tangent to the path

A comet orbits Sun counterclockwise as shown below. What is the direction of average velocity of the comet between points A and B?

Poll

A comet orbits Sun counterclockwise as shown below. What is the direction of intantaneous velocity of the comet between at point A?

Poll

Poll

A baseball is thrown with an x-velocity of 30 m/s and a y-velocity of -2 m/s. The ball is traveling

1. to the right and downward

2. to the left and downward

3. to the right and upward

4. to the left and upward

5. straight to the left

6. straight to the right

7. straight downward

8. straight upward

Speed is magnitude of instantaneous velocity

vx

vy

Poll

A high jumper leaves the ground with a speed of 8 m/s at an angle of 70 with respect to the ground. What is his x-velocity and y-velocity when he leaves the ground?

1. vx = 8 m/s, vy = 0 m/s

2. vx = 2.74 m/s, vy = 5.72 m/s

3. vx = 5.72 m/s, vy = 2.74 m/s

4. vx = 8 m/s, vy = 8 m/s

5. none of the above

Average Acceleration

As an object travels along a curve, its velocity vector will change in direction (and maybe in magnitude). The change in velocity per second is the average acceleration.

Poll

What direction is the average acceleration during the interval shown?

1.

2.

3.

Example

A football player is running with a velocity < 5, 5> m/s when he is hit by a defensive player. The collision lasts 1.5 s. His velocity after the collision is < 2, 0 > m/s. What is the player’s average acceleration during the collision?