Projectile motion graphs | Two-dimensional motion | AP Physics 1 | Khan Academy

Summary

This educational video script explores the physics of projectile motion on Earth, ignoring air resistance. It discusses the acceleration, velocity, and position in both the y (vertical) and x (horizontal) directions for three different scenarios: throwing a projectile straight out, at an angle downwards, and straight downwards. The script emphasizes that while acceleration due to gravity is constant and downward in the y direction, there is no acceleration in the x direction. The velocity and position graphs for each scenario are described, highlighting how to treat the x and y components independently.

Takeaways

• 🌍 In all scenarios, the acceleration in the y-direction is constant and downward due to gravity, represented as -g.
• 🚀 There is no acceleration in the x-direction for any scenario, assuming no air resistance.
• 📉 The velocity in the y-direction decreases at a constant rate due to gravity for all scenarios, regardless of the initial velocity vector.
• 🔄 The velocity in the x-direction remains constant for all scenarios since there is no acceleration in this direction.
• 🔵 For the first scenario, the projectile has a positive initial y-velocity, which decreases over time until it reaches zero at the peak of its trajectory.
• 🔵 In the x-direction for the first scenario, the projectile moves at a constant positive velocity, leading to a linear increase in position.
• 🔵 The blue scenario starts with zero initial y-velocity, accelerating downward over time, while the x-velocity is slightly higher than the first scenario.
• 🟡 In the third scenario, the projectile has a negative initial y-velocity, which continues to decrease, and the x-velocity is similar to the blue scenario.
• 📊 The position versus time graph for the y-direction shows an initial increase followed by a decrease, while in the x-direction, it shows a constant rate of increase.
• 📚 Once the initial velocity is broken down into x and y components, each component can be analyzed independently in terms of motion.

Q & A

What is the acceleration in the vertical (y) direction for a projectile launched on Earth?

The acceleration in the vertical direction is constant and due to gravity, which is directed downwards. It is represented as negative g, where g is the acceleration due to gravity.

Is there any acceleration in the horizontal (x) direction for a projectile on Earth, assuming no air resistance?

No, there is no acceleration in the horizontal direction. The acceleration remains at zero, assuming no air resistance.

How does the initial velocity vector of a projectile affect its acceleration due to gravity?

The direction in which the projectile is thrown does not affect the acceleration due to gravity. The acceleration due to gravity is always constant and directed downwards.

What is the initial velocity in the y direction for a projectile thrown straight out from a cliff?

The initial velocity in the y direction for a projectile thrown straight out is positive, as it is launched upwards.

How does the velocity in the y direction change over time for a projectile thrown straight out?

The velocity in the y direction starts positive and decreases at a constant rate due to the constant negative acceleration caused by gravity.

What is the initial velocity in the x direction for a projectile thrown straight out?

The initial velocity in the x direction is positive and remains constant over time, assuming no air resistance.

What happens to the velocity in the y direction for a projectile thrown at an angle downwards?

For a projectile thrown at an angle downwards, the initial velocity in the y direction is negative and becomes more negative over time due to gravity.

How does the velocity in the x direction compare between the scenarios of throwing the projectile straight out and at an angle downwards?

The velocity in the x direction remains constant for both scenarios, assuming no air resistance, but the magnitude may differ based on the initial throw.

What is the position-time graph like for a projectile thrown straight out in the y direction?

The position-time graph for a projectile thrown straight out in the y direction will show an initial increase followed by a decrease as the projectile reaches its peak and then falls back down.

How does the position in the x direction change for a projectile thrown straight out from a cliff?

The position in the x direction increases at a constant rate as the projectile moves horizontally with a constant velocity.

What is the key takeaway from analyzing the motion of a projectile in both the x and y directions?

The key takeaway is that once the initial velocity is broken down into x and y components, each component can be treated independently. The motion in one direction does not affect the motion in the other direction.

Outlines

00:00

🚀 Projectile Motion Analysis

The instructor discusses three different scenarios of projectile motion from the edge of a cliff on Earth. They consider the effect of gravity on the motion of the projectile, ignoring air resistance. The acceleration in the vertical (y) direction is constant due to gravity and is negative, indicating a downward force. In contrast, the horizontal (x) direction experiences no acceleration, meaning the initial velocity in x remains constant. The instructor breaks down the initial velocity vector into its x and y components and explains how these components change over time due to acceleration. They emphasize that the x and y components can be treated independently once separated.

05:02

📈 Graphing Projectile Motion

The instructor continues the analysis by discussing how to graph the position, velocity, and acceleration for each scenario. They explain that the initial position for all scenarios is the same, but the initial velocities differ, particularly in the y dimension. The instructor describes how the position graph for the first scenario (salmon colored) shows an initial positive y position that decreases until the velocity reaches zero, after which the position decreases. The x position increases at a constant rate due to the constant positive x velocity. The instructor also discusses the blue and yellow scenarios, explaining how their position and velocity graphs differ due to different initial velocities. They highlight the importance of understanding how to visualize and conceptualize these graphs to appreciate the independent behavior of the x and y dimensions in two-dimensional projectile motion.

Keywords

💡Projectile

A projectile is an object that is given an initial velocity and then moves along a curved path under the action of gravity alone. In the video, the instructor uses projectiles to demonstrate the principles of motion, specifically how they behave when launched from a cliff's edge with different initial velocities.

💡Initial Velocity Vector

The initial velocity vector is the vector representing the speed and direction of an object at the start of its motion. The video script describes how different scenarios involve launching projectiles with different initial velocity vectors, affecting their motion in both the x and y directions.

💡Acceleration

Acceleration is the rate of change of velocity of an object with respect to time. In the context of the video, the instructor discusses how the acceleration of a projectile is constant and due to gravity, represented as 'g' or '-9.81 m/s^2', indicating a downward force.

💡Velocity

Velocity is the speed of an object in a specific direction. The video explains how the velocity of a projectile changes over time in both the vertical (y) and horizontal (x) directions, starting with its initial velocity components and then being affected by gravity or lack thereof.

💡Position

Position refers to the location of an object in space relative to a reference point. The script describes how the position of a projectile changes over time, starting from the edge of a cliff and moving according to its initial velocity and the acceleration due to gravity.

💡Gravity

Gravity is the force that attracts two bodies towards each other, causing the projectile to accelerate towards the Earth. The instructor mentions ignoring air resistance to focus on the effect of gravity alone on the projectile's motion.

💡Air Resistance

Air resistance is the force that opposes the motion of an object through the air. The video script assumes no air resistance to simplify the analysis, focusing on the effects of gravity on projectile motion.

💡Laboratory Vacuum

A laboratory vacuum refers to an environment with little to no air, used to eliminate air resistance in experiments. The instructor mentions this concept to justify ignoring air resistance when analyzing the projectile's motion.

💡Deceleration

Deceleration is the reduction in velocity of an object. In the video, deceleration is discussed in the context of the vertical motion of the projectile, where the velocity decreases due to the constant acceleration of gravity.

💡Constant Acceleration

Constant acceleration means the acceleration does not change over time. The video script uses this concept to describe the motion of projectiles, where the only acceleration is due to gravity, which remains constant throughout the motion.

💡Two-Dimensional Projectile Motion

Two-dimensional projectile motion refers to the motion of an object that is projected into the air and moves under the influence of gravity, with its path described in two dimensions (x and y). The video script uses this concept to analyze the motion of projectiles thrown at different angles from a cliff's edge.

Highlights

Scenarios involve a person standing at the edge of a cliff on Earth launching a projectile in different directions.

The video will discuss acceleration, velocity, and position graphs for different initial velocity vectors in both y and x directions.

Acceleration in the y direction on Earth is constant and downward due to gravity.

Acceleration in the x direction is zero due to the absence of air resistance.

Velocity in the y direction starts positive and decreases at a constant rate due to gravity.

Velocity in the x direction remains constant if there is no air resistance.

The initial velocity vector can be broken down into its y and x components.

In the first scenario, the y velocity starts positive and decreases, while x velocity remains constant.

In the second scenario, the y velocity starts at zero and becomes more negative over time, while x velocity remains constant.

In the third scenario, the y velocity starts negative and becomes more negative, while x velocity remains constant.

Position graphs show different initial velocities in the y dimension affecting the trajectory.

The x position increases at a constant rate due to constant x velocity.

The y position graph shows an initial upward movement followed by a downward trajectory due to gravity.

The rate of acceleration is the same in all scenarios despite different starting points.

The x position in the second scenario has a slightly higher slope due to a higher initial x velocity.

The x position in the third scenario is similar to the second scenario due to similar x velocities.

The video emphasizes treating x and y dimensions independently once the initial velocity vectors are broken down.

Transcripts

- [Instructor] So in each of these pictures

we have a different scenario.

We have someone standing at the edge of a cliff on Earth,

and in this first scenario,

they are launching a projectile up into the air.

In this one they're just throwing it straight out.

They're not throwing it up or down but just straight out.

And here they're throwing the projectile

at an angle downwards.

And so what we're going to do in this video

is think about for each of these initial velocity vectors,

what would the acceleration versus time,

the velocity versus time,

and the position versus time graphs look like

in both the y and the x directions.

So I encourage you to pause this video

and think about it on your own or even take out some paper

and try to solve it before I work through it.

So let's first think about acceleration

in the vertical dimension,

acceleration in the y direction.

We're assuming we're on Earth and we're going

to ignore air resistance.

We can assume we're in some type of a laboratory vacuum

and this person had maybe an astronaut suit on

even though they're on Earth.

What would be the acceleration in the vertical direction?

Well the acceleration due to gravity will be downwards,

and it's going to be constant.

We're going to assume constant acceleration.

So the acceleration is going to look like this.

And if the magnitude of the acceleration

due to gravity is g,

we could call this negative g to show

that it is a downward acceleration.

Once the projectile is let loose,

that's the way it's going to be accelerated.

Now what about in the x direction?

Well if we assume no air resistance,

then there's not going to be any acceleration

or deceleration in the x direction.

So it's just going to be,

it's just going to stay right at zero

and it's not going to change.

And what I've just drawn here is going to be true

for all three of these scenarios

because the direction with which you throw it,

that doesn't somehow affect the acceleration due to gravity

once the ball is actually out of your hands.

So now let's think about velocity.

So what is going to be the velocity in the y direction

for this first scenario?

Well we could take our initial velocity vector

that has this velocity at an angle

and break it up into its y and x components.

So this would be its y component.

We just take the top part of this vector right over here,

the head of it, and go to the left,

and so that would be the magnitude of its y component,

and then this would be the magnitude of its x component.

So the y component, it starts positive,

so it's like that,

but remember our acceleration is a constant negative.

So our velocity is going to decrease at a constant rate.

So our velocity in this first scenario

is going to look something,

is going to look something like that.

Now what about the velocity in the x direction?

We see that it starts positive,

so it's going to start positive,

and if we're in a world with no air resistance,

well then it's just going to stay positive.

Notice we have zero acceleration,

so our velocity is just going to stay positive.

One of the things to really keep in mind

when we start doing two-dimensional projectile motion

like we're doing right over here

is once you break down your vectors into x and y components,

you can treat them completely independently.

That something will decelerate in the y direction,

but it doesn't mean that it's going to decelerate

in the x direction.

Now what would the velocities look like

for this blue scenario?

Well our velocity in our y direction,

we start off with no velocity in our y direction

so it's going to be right over here.

But then we are going to be accelerated downward,

so our velocity is going to get more and more

and more negative as time passes.

And notice the slope on these two lines are the same

because the rate of acceleration is the same,

even though you had a different starting point.

Now what about the velocity in the x direction here?

It looks like this x initial velocity

is a little bit more than this one,

so maybe it's a little bit higher,

but it stays constant once again.

Now let's look at this third scenario.

In this third scenario,

what is our y velocity, our initial y velocity?

Well it would look something like that.

And our initial x velocity would look something like that.

If we were to break things down into their components.

So our y velocity is starting negative,

is starting negative,

and then it's just going to get more and more negative

once the individual lets go of the ball.

Because you have that constant acceleration,

that negative acceleration,

so it's gonna look something like that.

And what about in the x direction?

Well looks like in the x direction right over here

is very similar to that one,

so it might look something like this.

I'll draw it slightly higher just so you can see it,

but once again the velocity x direction stays the same

because in all three scenarios,

you have zero acceleration in the x direction.

Now last but not least let's think about position.

So they all start in the exact same place

at both the x and y dimension,

but as we see, they all have different initial velocities,

at least in the y dimension.

So let's start with the salmon colored one.

So the salmon colored one,

it starts off with a some type of positive y position,

maybe based on the height of where the individual's hand is.

And then what's going to happen?

Well it's going to have positive but decreasing velocity

up until this point.

At this point its velocity is zero.

So its position is going to go up

but at ever decreasing rates until you get right

to that point right over there,

and then we see the velocity starts becoming more

and more and more and more negative.

So it would look something like,

something like that.

Now what would be the x position

of this first scenario?

Well if we make this position right over here zero,

then we would start our x position would start over here,

and since we have a constant positive x velocity,

our x position would just increase at a constant rate.

It would do something like that.

Now what about this blue scenario?

Well this blue scenario,

we are starting in the exact same place

as in our pink scenario,

and then our initial y velocity is zero,

and then it just gets more and more and more

and more negative.

So it would look something,

it would look something like this.

Now what about the x position?

Well our x position,

we had a slightly higher velocity,

at least the way that I drew it over here,

so we our x position would increase at a constant rate

and it would be a slightly higher constant rate.

So it would have a slightly higher slope

than we saw for the pink one.

Now the yellow scenario,

once again we're starting in the exact same place,

and here we're already starting with a negative velocity

and it's only gonna get more and more and more negative.

So it's just gonna do something like this.

It's gonna get more and more and more negative.

It's a little bit hard to see,

but it would do something like that.

And if the in the x direction,

our velocity is roughly the same as the blue scenario,

then our x position over time for the yellow one

is gonna look pretty pretty similar.

So this is just a way to visualize

how things would behave in terms of position,

velocity, and acceleration in the y and x directions

and to appreciate, one,

how to draw and visualize these graphs

and conceptualize them,

but also to appreciate that you can treat,

once you break your initial velocity vectors down,

you can treat the different dimensions,

the x and the y dimensions, independently.