hey everyone in this video we're going to learn about velocity in two-dimensional motion first we'll learn how to draw velocity vectors then we'll talk about two dimensional velocity vectors and how to convert between the magnitude and Direction and The X and Y components just like we did with displacement vectors next we'll talk about negative velocity components then we'll look at an example of adding velocity vectors and finding the magnitude and direction of the new velocity and finally we'll talk about how the X and Y velocity components are completely independent of each other and how we can take a two-dimensional problem and make it two one-dimensional problems so what's start with velocity vectors for most of this video we're going to use a bird's eye view of a curling Stone sliding on ice but first there's a few things we need to explain about the motion of the curling Stone ice doesn't have a lot of friction so things can slide pretty far in the real world there's still a little bit of friction so the curling Stone will eventually slow down and stop but in this video and in many physics problems we're going to assume that there's no friction that means once the stone is moving it's not going to slow down and it'll never stop on its own this is an example of Newton's first law of motion an object in motion will remain in motion at a constant speed in a straight line unless acted on by an unbalanced force we'll cover Newton's laws in a different section but this means that without friction the stone is going to move at a constant velocity a constant speed in a straight line unless some other Force gets involved so when we look at the motion of the curling Stone imagine it was pushed from somewhere out of view and now it has a constant velocity so the stone will move in a straight line at a constant speed all right now that we understand how the stone moves how do we describe its velocity using vectors let's say this Stone moves to the right at 2 m/s what would its velocity Vector look like let's focus on the initial point when the stone is here this is the velocity Vector at this time the vector points to the right which is the direction of the motion we can draw the vector on the object at a specific moment in time we can also draw the vector anywhere else if the picture gets too crowded we just need to keep track of the object and the time point that the vector represents so the velocity of the stone is 2 m/ second but how long should this vector be drawing a velocity Vector is a little different than drawing a displacement Vector displacement vectors are unique because displacement includes an initial and a final point so we could draw the vector connecting the points in 2D space but when we draw a velocity Vector we usually represent the instantaneous velocity of an object that means we're describing the magnitude and direction of the Velocity at one point one instant in time this also means that the velocity Vector is not tracing out the path of the object it's showing the direction of motion at one instant in time that'll make more sense later on so the magnitude of the Velocity is 2 m/s but we can draw the vector any length that we want having said that the length does matter if we're comparing multiple vectors for example if this stone is moving at 2 m per second and this one is moving at 4 m/s we draw a longer velocity Vector to represent a greater velocity if the velocity is double then the vector would be twice as long so if we're asked a question where there's multiple vectors in one picture just know that the length corresponds to the magnitude of the vector if they're velocity vectors A longer arrow means it's moving faster but most of the time if you're drawing vectors yourself just use numbers to keep track of the magnitudes and don't worry too much about drawing the vectors to scale so that's the velocity Vector at this initial Point what about 1 second later when the stone is here the velocity Vector at this time would be the same the stone has a constant velocity so the velocity Vector is the same at every point if the stone was moving in the y direction at 2 m/s then the velocity Vector would look like this and it would be the same at every point in time so those are examples of onedimensional velocity vectors now what if the stone is moving 2 m/s but it's moving in two-dimensional motion then the velocity Vector would look like this it points in the direction of the Velocity but if the stone is moving at 2 m/s in that direction how fast is the stone moving in the X Direction and how fast is it moving in the y direction is it moving to the right at 2 m/s and up at 2 m/s to answer that we need to find the X and Y components of the Velocity vector VX is the velocity component in the X Direction and VY is the component in the y direction we do the same thing that we did when we learned about displacement vectors and components the vector v and its two components VX and VY form a right triangle so we can use the trig functions to convert from the magnitude and Direction into the X and Y components but we need to be given an angle Theta in this case Theta is 60° s of theta equals the opposite side ID the hypotenuse v y is the opposite side from the angle and V is the hypotenuse if we rearrange this equation v y = v * the S of thet if we plug in 2 m/s for V and 60° for Theta we find that VY is 1.7 m/s as long as our calculator is set to degrees instead of radians we can do the same thing for the X component but using cosine instead of s because the X component is adjacent to the angle that we're using when we do that we find that VX is 1 m/s remember we use sign for the component that's opposite from the angle and cosine for the one that's adjacent to the angle if we were given the other angle then VX would use S and v y would use cosine so this stone is moving at 2 m/s at an angle of 60° but these components tell us that the stone is moving to the right at 1 m/s and up at 1.7 m/s this might seem confusing if your you're not familiar with velocity vectors using the right triangle trig functions to find the components of a displacement Vector makes sense because there side lengths of a triangle in 2D space but the same thing works for velocity vectors and velocity components if we look at the actual velocity of the stone and we ignore the X and Y components the stone is moving at 2 m/s in the direction of the velocity Vector but if we imagine the stone is casting a shadow onto the xais and we follow its X position over time then the velocity of this Shadow is 1 m/ second to the right over a period of 1 second the actual stone moves 2 m in the direction of its motion but the shadow moves 1 M along the xais this is just geometry the stone moved 1 m to the right in 1 second so its velocity to the right is 1 m/s if we follow the shadow on the Y AIS we see the shadow is moving upwards at 1.7 m/s the stone moves 1.7 m in the y direction in 1 second so a component of the Velocity Vector is like the velocity of the Shadow along one AIS that's how fast the object is moving purely in that direction let's do a quick recap by looking at some examples if the stone is moving directly to the right at 2 m per second then the X component of the Velocity is also 2 m/s and the Y velocity is zero if the stone is moving at 30° the x velocity is 1.7 m/s and the Y velocity is 1 m/s if the stone moves at 45° then both the X and Y velocity components are 1.4 m/s if the stone moves at 60° like before the x velocity vity is 1 m/s and the Y velocity is 1.7 m/s and if the stone moves directly upwards then the Y velocity is 2 m/s and the x velocity is zero so far we've only seen the velocity Vector point to the right and upwards but what if it points down or to the left in every physics scenario we need to establish the X and Y directions and which way is positive and negative in this example the positive X direction is to the right and the positive y direction is up this means that left is the negative X Direction and down is the negative y direction you can choose different directions depending on the situation but this is the most common way to set up the coordinate system just like we learned with displacement vectors a component can be positive or negative depending on its direction if the stone is moving in this direction VX is positive and VY is positive if the stone is moving in this direction VX is positive but v y is pointing in the negative y direction so VY is negative if the stone is moving in this direction then VX is negative and v y is positive and if the stone is moving in this direction VX and VY are both negative remember that we can draw the components in different places so do what makes the most sense to you you might have noticed that the components are switching between positive and negative but the 2 m/s didn't change a vector's magnitude is always positive 2 m/s is the speed of the stone which is the magnitude of the Velocity no matter which way the stone is moving we would still say its speed or the magnitude is positive 2 m/ second and again keep in mind that the components are positive or negative based on the direction that they're pointing not on the position of the object the positive and negative directions are always the same no matter where the object is it's like using Compass directions if a component is pointing to the right it's pointing in the East Direction which is positive if the component is pointing to the left it's pointing in the west Direction which is negative as a visual example here's a curling Stone moving in a circle it's its speed is constant so the velocity Vector is always the same length but watch how the X and Y velocity components change depending on which direction the stone is moving at any moment in time all right so we already learned how to take the magnitude and direction of the velocity and find the X and Y components now now let's do things in reverse and learn how to create a new velocity vector by adding components together here we have two stones that are not moving their X and Y velocities are zero Now Let's ignore the stones for a second and imagine the wind is blowing over one area of the ice and the wind's velocity is 1 m/s in the positive y direction in this example if a stone is in an area where there's wind then the wind pushes the stone and changes the Stone's velocity we're going to assume that the wind's velocity is added to the stone so the Stone's y velocity increases by 1 m/s if the stone had no velocity before its new y velocity would be a constant 1 m/s upwards the the wind is pushing the stone up so the wind increases the Stone's y velocity component Vy but it's not pushing the stone left or right so the wind does not change the X component VX which is still zero now let's imagine a new scenario where the stone is already moving to the right at 2 m/s before it's hit by the wind the X component is 2 m/s because the vector is pointing in the X Direction and the Y component is zero what will happen to the Stone's velocity when it's hit by the wind let's take a look the stone continues moving to the right but now it also moves up with the wind so the Stone's velocity is now two-dimensional it's moving in the X Direction and Y Direction at the same time and the new velocity Vector is in this diagonal Direction the wind increases the Stone's y velocity but the x velocity doesn't change it's still 2 m/s let's watch the Stone's velocity along the Y AIS it goes from 0 m/s to 1 m/s now let's look at the velocity along the X AIS the x velocity stays at 2 m/s the whole time so what is the magnitude and direction of the Stone's velocity after it's hit by the wind now that the stone has X and Y velocity components the actual velocity of the stone is the resultant Vector when we add the two components together this is just like when we added two displacement vectors graphically we can add vectors using the tip to tail method so if we draw VX and then we draw VY starting at the end of VX then the sum of the two components VX plus VY is this new vector v v starts at the beginning of the first component and ends at the end of the second component but what is is the value or the magnitude of this new vector v how fast is the stone moving after it's hit by the wind and what's the angle of the Velocity Vector earlier we used the magnitude and direction to find the X and Y components now we're going to use the components to find the magnitude and direction of the vector let's find the magnitude first for that we're going to use the Pythagorean theorem for a right triangle where C is the length of the hypotenuse and a and b are the other two sides then c^2 = a^2 + B2 if we apply that to this vector v is the hypotenuse C and VX and v y are the other sides A and B so V ^2 = vx2 + V y^2 we can take the square root of both sides so we just have V on the left then if we plug in 2 m/s for VX and 1 m/s for v y we find that V equals 2.2 m/s that's the length of the hypotenuse and it's the magnitude of the Velocity now let's find the vector's angle to do that we're going to use the inverse tangent function remember tan of theta is equal to the opposite side divid the adjacent side but if we want to finda Thena equals the inverse tent of opposite over adjacent if we're trying to find this angle then the opposite side is VY and the adjacent side is VX we plug in 1 m/s for VY and 2 m/s for VX and we find that Theta equal 26.6 de so at this moment in time when the stone is being pushed by the wind the stone is moving 2.2 m/s at an angle of 26.6 that's the actual magnitude and direction of the Stone's velocity and VX and VY are the components of that velocity in the X and Y directions so that's how we describe the velocity of an object in two-dimensional motion we can use the X and Y velocity components or we can use the magnitude and the angle and we can convert between them to answer different questions about the motion at this point there is an important concept we need to talk about which is a fundamental part of motion and forces and that's the fact that the X and Y motions are independent in the last example if the Stone's actual velocity was 2.2 m/s in this direction then why do we care about these things called components why is it useful to focus on the Stone's Motion in the X direction or the y direction here's the concept that's going to be very important as we study Physics an object's X motion and Y motion are independent of each other let's read that again in objects X motion and Y motion are independent of each other put another way X motion does not affect y motion and Y motion does not affect X motion this is a concept that you might not have thought about and it's not intuitive so it may take some time for it to sink in so what does this mean first motion can mean anything we've learned about in kinematics position displacement velocity or acceleration since we're covering velocity in this video let's use the word velocity how does this apply to our last example let's watch the motion of the stone again as it's hit by the wind the x velocity is 2 m/s to the right the entire time the wind does not change the Stone's x velocity because the wind doesn't push the stone in the X Direction the X motion and Y motion are independent so the change in the Y vol velocity has no effect on the x velocity the Y velocity changes because the wind pushes the stone in the y direction but the Y velocity has nothing to do with the fact that the stone is moving to the right the X motion does not affect the Y motion so by knowing that the X and Y velocities are independent we were able to take two separate velocity vectors and add them to find the Stone's new velocity the concept of the X and Y motions being independent will make more sense when we work with projectile motion but let's take a look at one more example using velocity here we have a boat crossing a river let's say the X direction is to the right along the river and the y direction is upwards across the river if the velocity of the boat is 10 m/s at an angle of 60° and the river is 40 m straight across how long does it take the boat to reach the other side of the river and how far does the boat move down the river to the right during that time keep in mind that for this example the water is not moving so we're given the boat's velocity as a vector with a magnitude and a Direction but we're being asked about the motion in the X Direction and the y direction separately this is where X and Y components are useful we'll skip the math for this example but if we use the magnitude the angle and the trig functions we can find the X and Y components of the Velocity Vector as the boat moves it's traveling 8.66 m/s upwards straight across the river and it's moving 5 m per second to the right along the river so now that we know the boat's velocity in each individual direction we can answer these questions first how long does it take the boat to reach the other side of the river we know the river is 40 m across which means the boat has to move 40 m in the y direction now that we know the boat's y velocity we can use that to find the time we can rearrange the equation for velocity so we get time equals the Y displacement / the Y velocity we plug in 40 M for Delta Y and 8.66 m/s for VY remember we're only focusing on the boat's y motion so we're using the Y displacement and the Y velocity this gives us 4.6 seconds for delta T which is how long it takes the boat to reach the other side of the river notice that the x velocity the velocity of the boat to the right has no effect on the time it takes across the river the only velocity that matters in this part is the velocity in the yde direction straight across the river instead of the Boat Moving 10 m/s at an angle we can imagine the boat is moving straight across the river at 8.66 m/s this is just like if the boat were casting a shadow along the Y AIS which we saw before with the curling Stone both of these boats have the same y velocity and they'll reach the other side in the same amount of time so that answers the first question now how far does the boat move down the river to the right during that time for this question we're looking at the motion in the X Direction however notice that the same period of time 4.6 seconds applies to the Y motion and the X motion the concept that time is shared between the X and Y motions will be important for projectile motion and we'll revisit that idea in another lesson so if the boat is moving to the right at 5 m/s how far does the boat move to the right in 4.6 seconds let's start with the equation for velocity in the X Direction and rearrange it so that Delta x is by itself on the left now we plug in 5 m/s for VX and 4.6 seconds for Delta Del T and we find that Delta X the displacement in the X direction is 23 M which is how far the boat moved down the river to the right during the time it took to reach the other side this is a basic example of why it's useful to know that the X and Y motions are independent we started with two-dimensional motion and we broke that down into one-dimensional motions then we used the atic equations to answer questions about the X motion and the Y motion separately so that's all we're going to cover in this video let's do a summary of the main points velocity Vector shows us the direction of an object's velocity at an instant in time so it represents the instantaneous velocity of the object the length of the vector represents the magnitude of the velocity which is the speed we can draw a velocity Vector any length but if we're comparing multiple vectors in one picture then the relative lengths should correspond to the relative magnitudes so if an object's velocity is constant then its velocity Vector should stay the same over time and if an object is moving twice as fast it should have a velocity Vector that's twice as long when an object is moving in two-dimensional motion that means its position is changing along the x axis and y AIS at the same time a 2d velocity Vector is at an angle and it has X and Y components the X component of the Velocity which we call VX is the velocity of the object in the X direction or we can think of it as the velocity of the object's Shadow along the X AIS and the same things are true for the Y component like with any Vector if we know the magnitude and angle we can use the cosine and S functions to find the X and Y components and if we know the components we can use the Pythagorean theorem and the inverse tangent function to find the magnitude and the angle once we establish the positive and negative X and Y directions a velocity component will be negative if it points in the negative Direction however the magnitude of the Velocity is always positive we can add velocity vectors just like we did with displacement vectors the resultant velocity Vector tells us the magnitude and direction of the object's actual velocity an important Concept in physics is that an object's X motion and Y motion are independent X motion does not affect y motion and Y motion does not affect X motion this means we can take two-dimensional motion and break that down into the X motion and the Y motion then we can study and answer questions about the X and Y motions separately that's it for this video thanks for watching and I'll see you in the next one