Kinematics Part 3: Projectile Motion

Summary

In this educational video, Professor Dave explores projectile motion, a concept where objects move both horizontally and vertically. He explains that the motion can be broken down into independent horizontal and vertical components, each with its own equations. Using the example of a rock thrown at a 30-degree angle from a cliff, he demonstrates how to calculate time in the air and horizontal distance traveled. The video simplifies the complex idea of projectile motion by relating it to one-dimensional motion, encouraging viewers to understand the principles through real-world application.

Takeaways

• 📚 Projectile motion involves an object moving in both horizontal and vertical directions, typically thrown or launched into the air.
• 🔍 Kinematic equations can be used to describe projectile motion by separating it into horizontal (x-direction) and vertical (y-direction) components.
• 🏋️‍♂️ Horizontal and vertical motions in projectile motion are independent, meaning they can be analyzed separately.
• 🎯 The path of a projectile is represented by a parabolic trajectory, influenced by the initial launch angle and velocity.
• 🕒 The time an object spends in the air during projectile motion is determined solely by its vertical motion.
• 📏 The horizontal distance traveled by a projectile is influenced by both its horizontal velocity and the time it is in the air.
• 📐 To analyze projectile motion, the initial velocity vector is split into its horizontal (Vx) and vertical (Vy) components using trigonometric functions.
• 📉 The vertical velocity of a projectile decreases due to gravity until it reaches the peak (zenith) and then becomes increasingly negative as it falls.
• ⏱️ The time a projectile is in the air can be calculated using the vertical motion equations, treating it as a one-dimensional vertical motion problem.
• 📐 The horizontal distance a projectile travels can be calculated by multiplying the horizontal velocity by the time the object is in the air.

Q & A

What is projectile motion?

Projectile motion is the motion of an object that is thrown or launched into the air, moving in both horizontal and vertical directions simultaneously. It can be represented by a parabolic path.

Why are the horizontal and vertical motions of a projectile considered independent?

The horizontal and vertical motions of a projectile are considered independent because they can be described by separate equations, with horizontal motion unaffected by gravity and vertical motion influenced only by gravity.

How does the time it takes for a projectile to hit the ground relate to its motion?

The time it takes for a projectile to hit the ground is determined solely by its vertical motion, as it will stop being in the air when it reaches the ground, regardless of its horizontal velocity.

What is the significance of the initial velocity's angle in projectile motion?

The angle of the initial velocity determines the direction and the distribution of the motion into horizontal and vertical components, which in turn affects the range and maximum height of the projectile.

How can the initial velocity vector of a projectile be split into horizontal and vertical components?

The initial velocity vector of a projectile can be split into horizontal (Vx) and vertical (Vy) components using trigonometric functions: Vx = V * cos(θ) and Vy = V * sin(θ), where V is the initial velocity and θ is the angle of projection.

What is the effect of gravity on the vertical velocity component of a projectile?

Gravity causes a constant acceleration in the negative direction on the vertical component of a projectile's velocity, causing it to decrease until it reaches zero at the peak (zenith) and then become increasingly negative until it hits the ground.

How can you calculate the time a projectile spends in the air?

The time a projectile spends in the air can be calculated using the vertical motion equation, considering only the vertical component of the initial velocity and the acceleration due to gravity.

What determines the horizontal distance a projectile travels?

The horizontal distance a projectile travels is determined by its horizontal velocity component and the time it spends in the air, as the horizontal motion is not affected by gravity.

Why are trigonometric functions necessary when analyzing projectile motion?

Trigonometric functions are necessary to break down the initial velocity into its horizontal and vertical components, which allows for separate analysis of the motion in each direction.

Can you provide a real-world example of how projectile motion is analyzed using the concepts from the script?

In the script, an example is given where a rock is thrown at a 30-degree angle with an initial velocity of 8.5 m/s from the edge of a 100-meter cliff. The analysis involves calculating the time in the air and the horizontal distance traveled using the principles of projectile motion.

Outlines

00:00

🚀 Understanding Projectile Motion

Professor Dave introduces the concept of projectile motion, explaining how it differs from horizontal and vertical motion. He uses the example of a cannonball fired at an angle to illustrate how the object's path can be represented by a parabola. The key takeaway is that the horizontal and vertical motions are independent, allowing for separate equations to describe each. The professor emphasizes this by comparing two marbles, one dropped and one projected horizontally, showing that they will hit the ground at the same time due to the independence of their motions. The discussion then transitions into a real-world example of throwing a rock off a cliff at a 30-degree angle with an initial velocity of 8.5 m/s, aiming to determine the time until it hits the ground and the horizontal distance it travels.

05:00

📚 Calculating Time and Distance in Projectile Motion

The second paragraph delves into the specifics of calculating the time a projectile spends in the air and the distance it travels horizontally. It explains that the time in the air is determined solely by the vertical motion, unaffected by horizontal velocity. The professor uses the kinematic equation to calculate the time the rock is in the air, finding it to be 4.97 seconds. With the horizontal velocity known, the distance traveled from the cliff's edge is then calculated by multiplying the horizontal velocity by the time in the air, resulting in approximately 36.6 meters. The paragraph concludes by reinforcing the idea that projectile motion, while more complex, can be analyzed by breaking it down into its horizontal and vertical components, each following one-dimensional motion principles.

Keywords

💡Projectile Motion

Projectile motion refers to the movement of an object through the air, influenced by gravity, where it has an initial velocity at an angle to the horizontal. In the video, the concept is central as it describes how objects like a cannonball follow a parabolic path when launched. The professor uses the example of a rock thrown off a cliff to illustrate how the motion can be broken down into horizontal and vertical components, emphasizing the independence of these components.

💡Kinematic Equations

Kinematic equations are formulas used in physics to describe the motion of an object in terms of its displacement, velocity, acceleration, and time. The video script mentions these equations as tools to describe both horizontal motion, like a car moving, and vertical motion, like an object in free fall. They are essential for analyzing projectile motion by separating it into horizontal and vertical components.

💡Horizontal Motion

Horizontal motion is the movement of an object along a plane parallel to the ground, unaffected by gravity (assuming air resistance is negligible). The script explains that for projectile motion, horizontal motion is independent of vertical motion. This is demonstrated when comparing the motion of two marbles dropped and rolled off a height, respectively; their horizontal motion does not affect their identical air times.

💡Vertical Motion

Vertical motion is the movement of an object along a line perpendicular to the ground, significantly influenced by gravity. In the video, vertical motion is discussed in the context of an object's fall, where the time it spends in the air is solely determined by this motion. The professor calculates the time a rock spends in the air using the vertical component of its velocity.

💡Independent Motion

The concept of independent motion in the script refers to the idea that an object's horizontal and vertical movements can be analyzed separately because they do not affect each other. This is crucial for understanding projectile motion, as it allows for the separate calculation of time in the air and horizontal distance traveled.

💡Parabolic Path

A parabolic path is the curved trajectory followed by an object in projectile motion under the influence of gravity. The script uses this term to describe the path of a cannonball fired at an angle, which travels upward before falling back down, creating a U-shaped curve. This shape is characteristic of the combination of horizontal motion at constant velocity and vertical motion under constant acceleration due to gravity.

💡Initial Velocity

Initial velocity is the speed of an object at the start of its motion and its direction relative to a reference frame. In the video, the professor calculates the initial velocity of a rock thrown off a cliff, breaking it down into horizontal (7.36 m/s) and vertical (4.25 m/s) components using trigonometric functions. This initial velocity determines the rock's trajectory and is critical for predicting its landing point and time.

💡Displacement

Displacement in the video refers to the change in position of an object. It is a vector quantity that considers both magnitude and direction. The script uses displacement (-100 meters) to calculate the time a rock is in the air when thrown vertically, illustrating how vertical displacement is influenced solely by gravity.

💡Acceleration Due to Gravity

Acceleration due to gravity is the constant acceleration that all objects experience when in free fall, directed downward towards the Earth. In the script, this acceleration is used to calculate the vertical motion of the rock, where it decreases until reaching zero at the peak (zenith) and then becomes increasingly negative as it falls back to the ground.

💡Trigonometric Functions

Trigonometric functions, specifically cosine and sine, are used in the script to break down the initial velocity vector of a projectile into its horizontal and vertical components. The professor uses these functions with the angle of launch (30 degrees) to calculate the initial velocities in the x and y directions, which are essential for analyzing the motion separately.

Highlights

Projectile motion involves an object moving in both horizontal and vertical directions simultaneously.

Kinematic equations can describe projectile motion by separating it into horizontal and vertical components.

Horizontal and vertical motions in projectile motion are independent of each other.

A parabola can represent the path of an object in projectile motion.

Objects with horizontal velocity and those dropped vertically will hit the ground at the same time if released simultaneously.

The horizontal motion of a projectile is unaffected by gravity, allowing for separate analysis.

The vertical motion of a projectile is solely influenced by gravity, ignoring air resistance.

The time a projectile spends in the air is determined by its vertical motion.

The distance a projectile travels horizontally is a function of both its horizontal velocity and time in the air.

The velocity of a projectile at any moment can be split into horizontal and vertical components.

Trigonometric functions are used to calculate the horizontal and vertical components of a projectile's velocity.

The initial horizontal velocity is calculated using the cosine of the launch angle.

The initial vertical velocity is determined by the sine of the launch angle.

The time a projectile is in the air can be calculated using the vertical motion equations.

The horizontal distance traveled by a projectile can be found by multiplying its horizontal velocity by the time in the air.

Projectile motion can be analyzed as a combination of two types of one-dimensional motion.

Understanding vector decomposition is key to analyzing projectile motion.

Transcripts

It's professor Dave, let's learn about

projectile motion.

We've learned how to use kinematic equations

to describe the motion of an object

moving horizontally, like a car, as well

as objects moving vertically, like

objects falling straight down to the

ground, but what about objects that move

in both of these directions? This type of

motion, if it involves an object that is

thrown or launched into the air, can be

referred to as a projectile motion.

Imagine a cannonball being fired at some

angle from the horizontal. It will travel

some distance up into the air before

eventually falling back down and hitting

the ground, some distance away from the

cannon, and we can use a parabola to

represent the path of this object.

The important thing to understand about

these kinds of examples is that the

horizontal motion and vertical motion of

the cannonball are completely

independent of one another.

This means we can use separate equations

to discuss the motion in each direction,

one equation that exclusively

corresponds to the x-coordinates of the

object and another that exclusively

corresponds to the y-coordinates of the

object. To drive this idea home consider

two marbles, one dropped from a

particular height and another that rolls

off of a surface at that same height

with some horizontal velocity. If these

begin falling at the same time they will

strike the ground at the same instant

because their vertical motion is

independent of any horizontal motion. The

one with horizontal velocity will cover

some distance in the X direction but it

will fall downward at the same rate as

the one that falls straight down and so

they will have identical airtimes. How

can we apply this to other real-world

examples?

well we have been throwing lots of rocks

off of cliffs lately so let's throw one

more. This time let's throw a rock at an

upward angle of 30 degrees off the

horizontal from the very edge of our

favorite 100-meter cliff and with an

initial velocity of

8.5 meters per second. Again we will ask

how long before the rock hits the ground?

and now additionally we want to know how

far away from the edge of the cliff it

will land. First things first let's make

sure we understand how these questions

relate to motion in both the x and y

direction. The time it spends in the air

only relates to Y direction behavior

because it will stop being in the air

when it hits the ground, no matter what

the horizontal velocity is, from 0 to

some huge number.

The vertical motion will be independent

of that. The distance it travels from the

edge of the cliff depends on the

horizontal velocity but also the time it

spends in the air because once it hits

the ground it can't travel any further.

And the velocity at any moment can also

be split into components. The horizontal

velocity will be the same at every

moment in this trajectory as long as we

disregard wind resistance, but the

vertical velocity will be the greatest

at the moment the rock is thrown and

then decrease until it reaches zero at

the zenith and then become increasingly

negative until it hits the ground.

This is because there is a constant

acceleration in the negative direction

due to gravity. We know we can use these

equations to answer these questions but

in order to discuss the x and y

directions

separately we have to split up this

velocity vector into x and y components.

This is pretty simple to do. We can

simply draw horizontal and vertical

vectors starting from the same point as

this one and extending precisely as far

as this one does in the X or Y direction.

These can be labeled V sub X and V sub Y

and we can think of these as being the

legs of the right triangle with the

existing vector as the hypotenuse. To

find out the magnitude of these vectors

we just do a little trig. We know that

the cosine of 30 will be equal to the

adjacent leg over the hypotenuse which

means 8.5 cosine 30 or 7.36 will be the

initial

velocity in the X direction.

We also know that the sine of 30 will be

equal to the opposite leg over the

hypotenuse so 8.5 sign 30 or 4.25

will be the initial velocity in

the y-direction. These velocities are

independent of one another, meaning that

a rock that was thrown straight up with

an initial velocity 4.25 m/s will land

on the ground at precisely the same time

as the one thrown at 8.5 meters per

second but at this 30 degree angle. That

means that if we want to figure out how

long it is in the air

this is no different than a problem with

one-dimensional vertical motion provided

that we understand that we are only

looking at the y component of this

velocity vector. So we can plug the

numbers in, -100 for the

displacement, 4.25 for initial velocity

and our usual acceleration. Putting it

into standard form and plugging into the

quadratic equation we will take the

positive result for t and say that the

rock will be in the air for 4.97 seconds.

This also makes it easy to calculate how

far the rock will travel, because we know

the horizontal velocity will be 7.36 m/s

times the 4.97 seconds that it is in the

air which leaves us about 36.6 meters

away from the edge of the cliff

So projectile motion is a little more

complicated than one-dimensional motion

but if we know how to divide any vector

into x and y components we can easily

analyze projectile motion as the

combination of two types of

one-dimensional motion. Let's check comprehension.

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