Bell Ringer:

• What is constant acceleration?

• What is projectile motion?

• Can we analyze the motion of an object on the y-axis independently of the object’s motion on the x-axis?

NOTES 3.2:

2D Motion: Projectile Motion

Physics Honors I

OBJECTIVES:

• Recognize that the vertical and horizontal motions of a projectile are independent.

• Relate the height, time in the air, and initial vertical velocity of a projectile using its vertical motion, and then determine the range using the horizontal motion.

• Explain how the trajectory of a projectile depends upon the frame of reference from which it is observed.

Active Physics Reference:

• Projectile Motion: Launching Things into the Air- Chapter 2, Section 4, Page 177-180

• The Range of Projectiles – Chapter 2, Section 5, Page 188-192

Further Learning:

• Khan Academy – Constant Acceleration:

https://www.khanacademy.org/science/physics/two-dimensional-motion

• Physics Classroom:

http://www.physicsclassroom.com/Class/vectors/u3l2a.cfm

• Physics – Principles and Problems – Red Book:

Chapter 6 – Section 1 – page 147

Sports Science – Projectile Motion:

Science of NFL – Projectile Motion:

Vectors in Two Dimensions:

• Last class, we analyzed displacement/distance in two dimensions through vectors.

• What we have not done so far is actually analyze the motion of the ball i.e. it’s velocity and acceleration.

• Like with displacement vectors, we can analyze the motion of an object on each axis INDEPENDENTLY of one another.

Vectors in Two Dimensions - Velocity:

• Written in unit vector notion so that:

v = vx Ƹi + vy Ƹj + vzk

v = (𝑣𝑖 cos 𝜃) Ƹi + (𝑣𝑖 sin 𝜃) Ƹj

Vectors in Two Dimensions - Velocity:

• Likewise, we can find the components of velocity vectors as we did with the displacement vectors:

𝑣𝑥 = 𝑣𝑖 cos 𝜃

𝑣𝑦 = 𝑣𝑖 sin 𝜃

Vectors in Two Dimensions - Velocity:

• Also as we did with the displacement vector, where we knew the components of the vector, then we could determine the magnitude and direction (angle) of the vector. Velocity can be analyzed in the same way such as:

Magnitude of Velocity Vector:

v = vx2 + vy

2 + vz2

Angle of a Velocity Vector:

θ = tan−1 ൗvy

vx

Vectors in Two Dimensions - Acceleration:

• As with displacement and velocity, we can greatly simplify problems by analyzing acceleration by its components along each axis. So if we write it in unit-vector notation, we get:

a = (𝑎𝑖 cos 𝜃) Ƹi + (𝑎𝑖 sin 𝜃) Ƹj

• Magnitude of Acceleration Vector:

a = 𝑎x2 + 𝑎y

2 + az2

• Angle of an Acceleration Vector:

θ = tan−1 ൗay

ax

a = ax Ƹi + ay Ƹj + azk

Vectors in Two Dimensions:

• The general point of all the above information is that physics problems can be solved easier by analyzing the problem based on what dimension it is in.

• Thus, try to break problems into analyzing the motion of a ball on the x-axis, the y-axis, and the z-axis.

• This is a very important concept when dealing with projectile motion problems and force problems.

Check Point 1:

1) An airplane is flying north with an airspeed of 200 km/h when it meets a crosswind of 70 km/h toward the east. Which of the following expressions determines the resultant speed of the airplane?

(a) (200 + 70) km/h

(b) (200 – 70) km/h

(c) 2002 + 702𝑘𝑚

ℎ

(d) 2002 − 702𝑘𝑚

ℎ

Check Point 2:

2) Alicia kicks a soccer ball with a velocity of 10 meters per second at a 60 degree angle relative to the ground. What is the horizontal component of the velocity?

3) The punter for the Jacksonville Jaguars kicks a football with an initial velocity of 18 meters per second at a 75-degree angle to the horizontal. What is the vertical component of the ball’s velocity at the zenith of its path?

What is Projectile MotioN?

• Seriously, what is it?

• Projectile Motion – is when something is projected or launched.

• Essentially, it is an object that after being thrown, projected, or launched, exhibits no other acceleration or forces acting on it except for gravity (because you are high-school students, we will neglect air resistance).

• Name an example of projectile motion?

• Name an example of something that is not projectile motion?

Let’s take a look at Projectile Motion?

• https://phet.colorado.edu/sims/html/projectile-motion/latest/projectile-motion_en.html

• Is the motion of an object launched in projectile motion dependent on it’s mass (after it’s launched)?

• What effect does changing the diameter of an object have on it?

• Does changing the diameter of the object effect it?

• Does drag force effect the range of the object being launched significantly?

• What about the velocity and the angle that the object is launched at?

Let’s take a look at Projectile Motion?

• https://phet.colorado.edu/sims/html/projectile-motion/latest/projectile-motion_en.html

(Click on Vectors)

• So we mentioned that the velocity on the x-axis is independent of the velocity on the y-axis. What does that mean?

• Look at what is happening with the velocity vector on the x-axis. What does that mean?

• What are we noticing about what is happening to the velocity vector on the y-axis. What does that mean?

Let’s take a look at Projectile Motion?

• On the x-axis, we notice that vector never changes. It stays a constant velocity throughout the entire process (neglecting air resistance).

Let’s take a look at Projectile Motion?

• On the y-axis, it goes up and it comes down. That is not different than me throwing this baseball straight up and coming down. It is the same as free-fall acceleration.

• The velocity on the y-axis does flip direction after reaching maximum height.

Projectile Motion - Independently:

• So what does this mean for us all:

• It means, we can break-up a problem of two-dimensional motion into its components:

Horizontal Motion – with zero accelerationVertical Motion – with constant downward acceleration

NOTICE: The horizontal component of velocity vector remains constant, while the vertical velocity component changes continuously.

Projectile Motion - Analyzed:

Let’s quickly analyze the motion of a penny being dropped versus a penny being projected vertically downward at an angle. What happens?

Projectile Motion – Horizontal Motion:

• Because there is no acceleration in the horizontal direction, the horizontal component 𝑣𝑥 of the projectile’s velocity remains unchanged from its initial value of 𝑣𝑥𝑖throughout the motion.

• So previously when we were analyzing one-dimensional motion with no acceleration, we used this formula:

∆𝑥 = 𝑣𝑥𝑖𝑡

• Because we have 𝑣𝑥𝑖 = 𝑣𝑖𝑐𝑜𝑠𝜃𝑖 which makes the equation become

∆𝑥 = (𝑣𝑖𝑐𝑜𝑠𝜃𝑖)𝑡

Projectile Motion – Vertical Motion:

• So vertical motion is exactly the same as it was when we threw a ball straight up and it came straight down. So basically, we just use the free-fall acceleration equations.

𝑣𝑦𝑓 = visin𝜃𝑖 + gt

v𝑦𝑓2 = (𝑣𝑖sin𝜃𝑖)

2 + 2gh

h = (visin𝜃𝑖)t +1

2gt2

h = v𝑦𝑓t −1

2gt2

h =1

2v𝑖 sin 𝜃𝑖 + v𝑦𝑓 t

Projectile Motion – Vertical Motion:

• As when a ball is thrown vertically upward, it is directed upward initially and its magnitude steadily decreases to zero, which marks the maximum height of the path. Then ball changes direction and begins to start coming down.

• Note: Vertical motion and horizontal motion are connected through the variable of time.

• The time from the launch of the projectile to the time it hits the target is the same for both vertical motion and horizontal motion. Therefore, solving for time in one of the dimensions, vertical or horizontal, automatically gives you time for the other dimension.

Projectile Motion – Trajectory:

• The entire motion of a projectile can be described by the parabolic formula(𝑦 = 𝑎𝑥 + 𝑏𝑥2) known as the Equation of the Path or Trajectory:

y = (tan θi)x −gx2

2(vi cos θ)2

• The Trajectory equation can be found by combining the following equations:

𝑥𝑓 − x0 = v𝑖 cos θ𝑖 t

h = (vi sin θ𝑖)t +1

2gt2

Projectile Motion – HORIZONTAL RANGE:

The horizontal range R of the projectile is the horizontal distance the projectile has traveled when it returns to its initial (launch) height.

𝑅 = 𝑥𝑓 − 𝑥𝑖 = 𝑣𝑖𝑐𝑜𝑠𝜃𝑖 𝑡

𝑅 =𝑣𝑖2

𝑔sin(2𝜃𝑖)

• Where sin 2θi = 2 sin θi cos θi

sin θi cos θi =1

2sin 2𝑖

Projectile Motion – HORIZONTAL RANGE:

The horizontal range R of the projectile is the horizontal distance the projectile has traveled when it returns to its initial (launch) height.

𝑅 = 𝑥𝑓 − 𝑥𝑖 = 𝑣𝑖𝑐𝑜𝑠𝜃𝑖 𝑡

𝑅 =𝑣𝑖2

𝑔sin(2𝜃𝑖)

• IMPORTANT: The horizontal range R is maximum for a launch angle of 45°.

• THE ABOVE EQUATION DOES NOT GIVE THE HORIZONTAL DISTANCE TRAVELED BY A PROJECTILE WHEN THE FINAL HEIGHT IS NOT THE LAUNCH HEIGHT.

Projectile Motion – HORIZONTAL RANGE:

The diagram below shows plots of trajectories launched at many different angles, but always with the same initial speed.

Example 1:

A stone is thrown horizontally at a speed of 5.0 m/s from the top of a cliff that is 78.4 m high.

(a) How long does it take the stone to reach the bottom of the cliff?

(b) How far from the base of the cliff does the stone hit the ground?

(c) What are the horizontal and vertical components of the stone’s velocity just before it hits the ground?

Example 2:

Courtney kicks a soccer ball at rest on level ground giving it an initial velocity of 7.8 m/s at an angle of 32°.

(a) How long will the ball be in the air?

(b) How high will the ball go?

(c) What will be its range?

Check Point 2:

4) A cannonball fired at a 20-degree angle to the horizontal travels with a speed of 25 meters per second. How many meters away does the cannonball land if it falls to the ground at the same height from which it launched.

5) A ball is launched at 4.5 m/s at 66° above the horizontal. What are the maximum height and flight time of the ball?

Just for your KnowledgeTerminal Velocity:

Just for your KnowledgeTerminal Velocity:

Terminal Speed (𝐯𝐭) – when a body falls at a constant speed that means it is no longer accelerating ( Ԧ𝑎 = 0).

𝐷 − 𝐹𝑔 = 𝑚𝑎

C = Drag coefficientρ = density A = the effective cross-sectional area of the body (The area of a cross-section taken perpendicular to the velocity v.NOTE: Drag force is similar to a frictional force for liquids or gases.

Exit Ticket 1:

5) Which of the following is an example of projectile motion?

(a) a jet lifting off a runway(b) a thrown baseball (c) an aluminum can dropped straight down into the recycling bin(d) a space shuttle being launched

6) What is the path of a projectile (in the absence of friction)?

(a) A wavy line(b) A parabola(c) A hyperbola(d) Projectiles do not follow a predictable path.

Exit Ticket 1:

7) If a pen and a ruler are dropped together from the same height, will they reach the ground at the same time? Explain your answer.

8) When an object falls vertically down, does its velocity remain the same? Explain your answer.

9) If a ball is thrown upward, what is the ball’s velocity at its point of highest rise? What is the ball’s acceleration?

10) If the launching and landing heights for a projectile are equal, what angle produces the greatest range? Why?