Two Dimensional Motion

Physics

Maple Hill High School 2025-26

Contents:

Review: Vectors

Vector Quantities:

A vector is a quantity with both magnitude (size) and direction.

Examples:

• The child was displaced 9 meters North.
• The car has a velocity of 10 meters per second East
• , , , , , etc

Scalar Quantities

A scalar is a quantity with just magnitude.

Examples

• The child traveled a distance of 12 meters
• The car is moving 20 miles per hour
• The frog has mass of 0.5 kg.
• , , , etc

Representing Vectors

Representing Vectors

Vector Direction Conventions

1. expressed as an angle of rotation about its "tail"
   • 40 degrees North of West (meaning a vector pointing West has been rotated 40 degrees towards the northerly direction)
   • 65 degrees East of South (meaning a vector pointing South has been rotated 65 degrees towards the easterly direction).
2. expressed as a counterclockwise angle of rotation about its "tail"
   • 30 degrees is a vector that has been rotated 30 degrees in a counterclockwise direction relative to due east
   • 160 degrees is a vector that has been rotated 160 degrees in a counterclockwise direction relative to due east

Vector Direction Conventions

The Magnitude of a Vector

• Magnitude of a vector is scaled in the vector diagram by the length of the arrow
• This can be scaled (i.e. 1 cm = 5 miles)

Adding Vectors

• When vectors are in the same, or opposite directions we can add them with regular algebra

Try It...

Mr. Porter and his wife walk from Nighthawks to Troy Savings Bank Music Hall. They walk 6 blocks East and then 2 Blocks South. [1 Block = 100 Meters]

1. Determine the distance that they traveled.
2. Determine their displacement.

Adding Angled Vectors

Adding Vectors

Vectors are added "tip to tail", that is redraw the vectors so that the tip of one vector is attached to the tail of the second vector.

The resultant vector S is equal to the addition of vectors a and b

Right Angled Vectors

Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric's resulting displacement.

Determinging the Angle

The Calculated Angle isn't always the answer...

Using Scaled Vectors

Scaled Vectors

Scaled Vectors

Resultant: the vector sum of two or more vectors. It is the result of adding two or more vectors

• Resultant Displacement or Resultant Force or Resultant Velocity

Order Does Not Matter

Stop

Representing Vector Components

Vector Components

• Parts of a two-dimensional vector
• The component of a vector is the influence of that vector in a given direction.
  • i.e. How far East of a North East displacement did you walk?
• We look at the perpendicular components
  • How much of the vector is in the x-direction
  • How much of the vector is in the y-direction
• Vector is made up of components and

Vector Components Math

What are the components of Vector ?

What are the components of Vector and ?

Use Trig to Find and

Adding Vectors

Using Components

You can sum the components of the two vectors to find the components of the resultant vector

If ...

Adding Perpendicular Vectors

Mac and Tosh are doing the Vector Walk Lab. Starting at the door of their physics classroom, they walk 2.0 meters, south. They make a right hand turn and walk 16.0 meters, west. They turn right again and walk 24.0 meters, north. They then turn left and walk 36.0 meters, west. What is the magnitude of their overall displacement?

A graphical representation of the given problem will help visualize what is happening. The diagram below depicts such a representation.

Step 1: Tip-to-tail

Step 2: Sum and directions

Practice

Adding 2 Perpendicular Vectors

Adding 3 or More Perpendicular Vectors

Vector Treasure Hunt

1. Create a Vector Map (draw this on the map of the school) -> start at one of the entrances of the school.
2. Record all vectors in component form and Magnitude-direction from (use the convention degrees clockwise from North...this is the setting your iPhone compass uses)
3. Exchange vector directions with another group and draw their vector map on your grid in another color or symbol (i.e. dotted line)
4. Calculate the displacement of the the vector map
5. Go walk the map and record the landmarks at the end of each vector.

Vector Descriptions

• Component Form:
• Magnitude-direction form:
  • units at degrees

Relative Motion

Relative Velocity and Vector Addition

• Goal: Understand how velocities add as vectors and why motion is relative to the observer.

Complete AP Workbook 1.E

1D Relative Motion: Vector Sums

Pivot: One-Dimensional Relative Motion Using a Drone

Boat Crossing a River: Resultant Velocity

When a boat heads straight across, the river current carries it downstream. The boat’s velocity relative to water adds vectorially to the current’s velocity relative to ground.

Pivot: 2D Motion with a Drone

Free Fall Motion

Free Fall

An object in free fall experiences only the force of gravity.

• This means that is accelerates at which is referred to as "acceleration due to gravity" or "gravitational field strength
• Units N/kg and m/s/s are the same

Free Fall Facts

→ Always when near the surface of the Earth

Free Fall Facts

If object is dropped

Free Fall Facts

If object is projected upwards it will slow down as it rises. The y-velocity will be zero at its heights point or peak

You can treat at the peak

Free Fall Facts

If an object is projected upwards the velocity at which it is projected is equal and magnitude and opposite in direction when it returns to its initial height.

i.e. a ball projected upwards at will have a downward velocity of when it returns to its launch height

Problem-Solving Process

1. Draw a diagram/picture of physical situation
2. Sketch Physics motion graphs (XT & VT)
3. Fill out cross diagram with givens and unknowns
   • (remember for all free fall questions)
4. Identify equation with cross diagram
5. Substitute and solve

Free Fall Kinematics

1. A ball is thrown downward with an initial speed of 20 m/s on Earth.
   1. Calculate the displacement of the ball during the first 4 seconds.
   2. Calculate the time required to reach 50 m/s.
   3. Calculate the time required to reach 50 m/s.
   4. Calculate the speed after falling 100 meters.

Free Fall Kinematics

2. A rock is thrown upward with an initial speed of 15 m/s on Earth.
   1. Calculate the rock's height after 1 second
   2. Calculate the time required for the rock to reach an upward speed of 3 m/s.
   3. Calculate the time required for the rock to reach a downward speed of 5 m/s

Free Fall - Calculator Free

Free Fall 1

Free Fall Kinematics

3. NASA operates a drop tower in which they test the response of materials to situations in which the only force which acts upon them is gravity. Objects are dropped from rest and free fall for 5.27 seconds through the drop tower.
   1. Determine the distancw which the objects fall through the tower.

Free Fall - Dropped Objects

Free Fall Kinematics

4. During a parachuting mishap (it could be worse), a parachutist who is falling at 12.2 m/s drops his new camera from an altitude of 78.9 m. What speed will the camera have when it strikes the ground?
5. A ball is thrown vertically upwards with a speed of 44.5 m/s.
   1. How high does the it rise above the point of release?
   2. How much time does it take the ball to reach the peak of its trajectory?

Additional Free Fall Practice

Free Fall 3

Free Fall 4

Rocket Science

Horizontal Projectiles

Two toy trucks roll off the ends of identical tables. The speeds and masses of the trucks are given.

Will Truck A be in the air for (i) a longer time, (ii) a shorter time, or (iii) the same time as Truck B before it reaches the floor?

Explain your reasoning.

Two toy trucks roll off the ends of identical tables. The speeds and masses of the trucks are given.

Will Truck A be in the air for (i) a longer time, (ii) a shorter time, or (iii) the same time as Truck B before it reaches the floor?

Explain your reasoning.

Launcher Demo

Horizontal Projectiles

• A projectile is an object that only experiences a gravitational force
• For horizontal projectiles the
• We can separate motion in the and planes and solve separately

Path of a Projectile

Path of a Projectile

Horizontal and Vertical Velocity

Horizontal and Vertical displacement

Horizontally Launched Projectiles

• Projectile launched with an initial horizontal velocity from an elevated position.
• Predictable unknowns include the initial speed of the projectile, the initial height of the projectile, the time of flight, and the horizontal distance of the projectile.

Horizontal Projectils

Problem-solving Process

1. Read & sketch problem
2. Identify given information & fill out double cross diagram
3. Identify the quantity you are solving for
4. Using either horizontal or vertical information, find the flight time of the projectile
5. With the time find the quantity you are solving for.

Example

A pool ball leaves a 0.60-meter high table with an initial horizontal velocity of 2.4 m/s. Predict the time required for the pool ball to fall to the ground and the horizontal distance between the table's edge and the ball's landing location.

Practice

A soccer ball is kicked horizontally off a 22.0-meter high hill and lands a distance of 35.0 meters from the edge of the hill. Determine the initial horizontal velocity of the soccer ball.

Practice

In many locations, old abandoned stone quarries have become filled with water once excavating has been completed. While standing on a quarry wall, a boy tosses a piece of granite into the water below. If he throws the rock horizontally with a velocity of 3.0 m/s, and it strikes the water 4.5 meters away, how high above the water is the wall?

Practice w/ Table

Suppose that an airplane flying 60 m/s, at a height of 300 meters, dropped a sack of flour. How far from the point of release would the sack have traveled when it struck the ground? Where will the plane be in relation to the sack when it hits the ground?

Practice

A stone is thrown horizontally to the right at a speed of 17.2 m/s from the top of a cliff that is 91.4 m high. Consider up and to the right positive directions.

1. How long does it take the stone to reach the bottom of the cliff?
2. How far from the base of the cliff does the stone hit the ground?
3. What is the horizontal component of the stone's velocity just before it hits the ground?
4. What is the vertical component of the stone's velocity just before it hits the ground?

Marble Mini-Experiment

1. Determine the launch velocity of your marble.

Equipment:

• Ruler track
• Marble
• Carbon paper
• Meterstick

2. Use to predict landing spot from new height (like off top of cabinets)

Cannonballs of different masses are shot from cannons at various angles above the horizontal. The velocity of each cannonball as it leaves the cannon is given, along with the horizontal component of that velocity, which is the same.

Rank the horizontal distance traveled by the cannonballs.

Cannonballs with different masses are shot from cannons at various angles above the horizontal. The velocity of each cannonball as it leaves the cannon is given, along with the same vertical component of that velocity.

Rank the time the cannonballs are in the air.

Components

• The X-axis velocity component () is constant and will not accelerated so it will stay the same the entire flight
• The initial Y-axis velocity component will change as it accelerated by gravity 9.8 m/s² down. This is why we say () and not just because it will change throughout the problem.

Finding x & y components of initial velocity

Problem-solving Process

1. Read & sketch problem
2. Calculate x & y components of initial velocity
3. Identify given information & fill out double cross diagram
4. Identify the quantity you are solving for
5. Using either horizontal or vertical information, find the flight time of the projectile
6. With the time find the quantity you are solving for.

Projectile Fact Reminders

Some Reminders from Free Fall

• → Always when near the surface of the Earth
• If object is dropped
• If object is projected upwards it will slow down as it rises. The y-velocity will be zero at its heights point or peak
  • You can treat at the peak
• If an object is projected upwards the velocity at which it is projected is equal and magnitude and opposite in direction when it returns to its initial height.
  • i.e. a ball projected upwards at will have a downward velocity of when it returns to its launch height

Where should go?

Riverboat Problems

• River current is 4 m/s North
• Boat Velocity 4 4 m/s
• What does a drone see?
• How can we predict this resultant velocity?

Example Riverboat

A motorboat traveling 4 m/s, East encounters a current traveling 3.0 m/s, North.

1. What is the resultant velocity of the motorboat?
2. If the width of the river is 80 meters wide, then how much time does it take the boat to travel shore to shore?
3. What distance downstream does the boat reach the opposite shore?

Riverboat Practice