3.5 Addition of Velocities

• Projectile to the satellite.
• A projectile is launched from a very high tower to avoid air resistance.
• The range becomes longer because the Earth curves away when initial speed increases.
• A large enough initial speed is all that is needed.

• Shoot various objects to learn about projectile motion.
• Try to hit a target in the simulation.

• The boat does not move in a certain direction.
• The river carries the boat downstream.
• The movement of the air mass relative to the ground carries the plane sideways.

• A boat going across a river will move in a different direction relative to the shore.

• An airplane is carried to the west and slowed down by the wind.
• The plane does not move relative to the ground, but rather in the direction of its total velocity.

• It is useful to add velocities in these two situations.
• In this module, we first look at how to add velocities and then look at some aspects of relative velocity.

• The addition of velocities is simple in one-dimensional motion.
• If a field hockey player is moving straight toward the goal and drives the ball in the same direction with a velocity of relative to her body, then the goalkeeper is in front of the goal.

• In two-dimensional motion, either graphical or analytical techniques can be used.
• Analytical techniques will be the focus.

• The sum of component vectors and is thevelocity of an object traveling at an angle to the horizontal axis.

• These equations can be used for any vectors.
• When the magnitude and direction of the velocity are known, the first two equations are used to find the components.
• When its components are known, the last two are used to find the magnitude and direction of the velocity.

• There is a toy boat in the water.
• Water starts to drain when the drain is unplugged.
• You can push the boat from one side of the tub to the other.
• The directions of the water, the heading of the boat, and the actual speed of the boat can be compared.

• A boat is attempting to travel across the river at a speed of 0.75 m/s.
• The current in the river is 1.20 m/s to the right.

• Figure 3.46 shows a boat trying to cross the river.
• The magnitude and direction of the boat's speed can be calculated.
• The boat's speed is 0.75 m/s in the -direction relative to the river and 1.20 m/s to the right.

• The boat's speed relative to the water is parallel to the river's speed.

• The river is swept quickly because it is large compared to the boat's speed.
• The small angle shows that the total velocity has a relation to the riverbank.

• The plane is moving at 45.0 m/s due north and 38.0 m/s west of the ground.

• An airplane is heading north at 45.0 m/s, but it has a speed of 38.0 m/s at an angle west of north.

• We know the total velocity and that it is the sum of two other velocities, the wind and the plane relative to the air mass, which is different from the previous example.
• The quantity is known and we are asked to find it.

• The components of the velocities can be found along a common set of axes.

• If we can find the components, we can combine them to solve the problem.

• The x- and y-components of the wind and plane velocities are the sums of the x- and y-components.

• The diagram shows motion west which is consistent with the minus sign.

• The diagram shows motion south which is consistent with the minus sign.

• The magnitude and direction of the wind are now known thanks to the components of the wind velocity.

• Because the plane is fighting a strong combination of crosswind and head-wind, it ends up with a total velocity significantly less than its velocity relative to the air mass as well as heading in a different direction.

• We were able to make the mathematics easier by choosing a coordinate system with one axis parallel to one of the velocities.
• We will find that choosing an appropriate coordinate system makes it easier to solve a problem.
• In projectile motion, we use a coordinate system with one axis parallel to gravity.

• We have been careful to say that the velocity is relative to the reference frame.
• The velocity of an airplane relative to an air mass is different than it is relative to the ground.
• It's not the same as the airplane's speed relative to its passengers, which should be close to zero.

• Albert Einstein, the greatest physicist of the 20th century, is associated with relativity by nearly everyone.
• We will study Einstein's modern theory of relativity in later chapters.
• The relative velocities in this section were first discussed by Galileo and Einstein.
• This speed is slower than most things we encounter in daily life.

• An example of what two different observers see in a situation was analyzed by Galileo.
• A sailor on a moving ship drops his binoculars.
• If air resistance is not very high, the binoculars will hit at the base of the mast directly below its point of release.
• Let's take a look at what two different people see when the binoculars drop.
• There are two people on the shore and one on the ship.
• The observer on the ship can't see the binoculars because they don't have a horizontal velocity.
• The binoculars and the ship have the same horizontal velocity, so they both move the same distance.
• The binoculars hit at the base of the mast, not behind it, because the paths look different to the different observers.
• It is important to specify the velocities relative to the observer in order to get the correct description.

• The same motion is seen by two different people.
• The binoculars dropped from the top of the mast fell straight down as an observer watched.
• The binoculars move forward with the ship as an observer on shore watches.
• The deck at the base of the mast has binoculars on it.
• The initial horizontal velocity is different from the two other observers.

• The plane is moving at 260 m/s and a passenger drops a coin.

• The motion of a coin was seen by two different people.

• Both problems can be solved with falling objects and projectiles.

• This motion is a projectile motion because the initial velocity is 260 m/s horizontal relative to the Earth and gravity is vertical.
• It is best to use a coordinate system with both horizontal and vertical axes.

• The square root of 29.4 has two roots.
• We chose the negative root because we know that the direction of the velocity is downwards, and we have a plan for the positive direction to be upwards.
• The motion is straight down relative to the plane because there is no initial horizontal velocity or horizontal acceleration.

• The final vertical velocity for the coin relative to the ground is the same as found in part (a), because the initial vertical velocity is zero relative to the ground and vertical motion is independent of horizontal motion.
• There is a horizontal component to the velocity.
• The initial and final horizontal velocities are the same.

• The final velocity is the same as it would be if the coin were dropped from the plane.
• In moving cars, this result is also true.
• An observer on the ground sees a different motion for the coin.
• The final velocity of the plane is barely greater than the initial one.

• In two dimensions, the final velocity v in part (b) is not the same as ordinary numbers.
• To see the difference between the airplane and the velocity, the magnitude had to be five digits.
• The motions of the plane and the ground are very similar, except that the plane's speed is much larger than that of the ship, so that the two observers see very different paths.
• One observer sees the coin fall vertically, but the other sees it move forward on the ground.
• One observer sees a vertical path while the other sees a horizontal path.

• The outcomes are spectacularly unexpected because Einstein was able to clearly define how measurements are made and because the speed of light is the same for all observers.
• Energy is stored as mass increases, and more surprises await.

• Try the latest version of the "Motion in 2D" simulation.
• You can learn about position, velocity, and acceleration.
• The simulation can move the ball in four different types of motion.
• You can open media in a new browser.

• In two dimensions, this path can be represented by drawing the first vector on a graph and then placing it with horizontal and vertical components.

• The result is drawn from one another.

• The direction of the product points in the same direction if it is positive and in the opposite direction if it is negative.

• The head-to-tail method is used.
• Subtraction is defined as Analytical Methods.
• The Pythagorean theorem protractor is used to determine the magnitude and direction of the addition and subtraction.

• The head-to-tail method of addition is followed in the first step.

• 3.4 Projectile Motion on level ground launched at an angle above the horizontal with initial speed is given.

• The following steps are used to solve projectile motion problems.
• A coordinate system can be determined.
• The velocities in two dimensions are resolved using the same position and/or velocity of the object in the horizontal analytical vector techniques, which are rewritten as and vertical components.

• The motion of the projectile in the horizontal can be analyzed using the following equations.

• The motion of the projectile in the vertical observers can be analyzed.

• Subtraction: Graphical Methods units, and direction are what you can give a specific example of.

• Subtraction: Analytical Methods illustrated below show how the same spot on a lake is different from the other one.
• The total distance between Path 1 and Path 2 is 7.5 km.

If you add two vectors, what do you think?

• Give an example of a non zero vector.

• The range of a projectile is determined by the angle at which it is fired.
• There are two angles that give the same range.
• The smaller angle is preferable because of factors that might affect the archer's ability to hit a target.

Are you able to end up from the other over the edge?

• The hat of a jogger that falls off keeps him focused on the players around him.
• The back of his head is where he is.
• The path of moving fast is shown in the sketch.
• He doesn't need to keep an eye on the jogger's hat.

• The bed of the truck is covered in dirt.
• Is it possible for the softball to hit the ground directly below the end of the truck?

• Suppose you walk 18.0 m west and then 25.0 m north.
• These problems can be solved using graphical methods.

• Different people are walking in a city.
• The blocks are on the side.

• The two displacements add up to give a total displacement with direction and magnitude.

• If you first walk 12.0 m in a direction west of the displacement, you'll have a good idea of how far you'll go.

• The same final Subtraction: Analytical Methods result can be seen if you add the two legs of the walk.

• Different people walking in a city are represented by the various lines.
• The blocks are on the road.
• Find the sum and side at the same time.

• The total is given by the two velocities and add.

• Figure 3.55 shows the components of a set of axes.

• A new owner has a triangular piece of land and she wants to fence it.
• She started at the west corner.
• The two legs of the walk are represented by you as the third side is calculated.

• The two displacements add up to give a total displacement with direction and magnitude.

• In the direction south of west, you fly in a straight line.

• You can also solve it graphically.
• The analytical technique for solving this problem is to arrive at the same point if you fly straight south and west.

• If you find the distances you would have to fly first, you will get the same result.

• Discuss how taking displacement along a different set of axes might help to reach the same point.

• You drive in a straight line.
• The first three sides are shown east of north.
• What is the same point for him?

• The wind shifts a lot during the over a line of buses parked end to end by driving up a day, and he is blown along the following straight lines: ramp at a speed.

• The bull's-eye of the target is the same height as the release from the starting point and the direction of the arrow.

• In this part of the problem, show how you from the north and how the wind would affect the steps you take to solve the plane problems.

• A rugby player passes the ball across the field, where it is caught at the same height as it left his hand.

• A maximum speed of 50.0 m/s can be achieved by the cannon on the battleship.
• The shell strikes a target above the ground.

Does your answer imply that the error introduced by the building and lands 100.0 m from the base of the assumption of a flat Earth is building?

• The ball is shot at a angle of 30 m/s and hit above the horizontal.

• Can a goalkeeper kick a soccer ball into off with the legs to see how far one can jump?
• The distance will be about 95 m.

• You should state your assumptions.

• The world long jump record is held by Mike Powell of the USA.
• You should state your assumptions.

• A tennis player hits the margin of error while serving at a speed of 170 km/h.
• Show how you follow the ball at a height of 2.5 m and an angle below the steps involved in projectile motion problems.

• In 2007, Michael Carter set a world record.

• The maximum distance for a projectile on speed of 2.00 m/s when a quarterback throws a pass to a player is achieved when air resistance is straight downfield.

• He keeps his horizontal speed.

• A football player punts the ball.

• An eagle is flying at a speed of 3.00 m/s with a brief gust of wind that reduces its horizontal speed when the fish in her talons wiggles loose and falls into by 1.50 m/s.
• What is the distance from the lake to the ball?

• An owl is carrying a mouse.
• The form is there.
• To get this position at that time, you have to solve the equation for and center of the 30.0 cm nest.
• The owl is flying at 3.50 m/s at an angle below the horizontal when it accidentally drops the mouse.
• The horizontal position of the mouse constants can be calculated to answer the equation of the form where and are question.

• A soccer player kicks the ball from a long way away.
• To find the initial speed of the ball, you need to find the time at which it passes over the goal, 2.4 m above the ground, given zero, and substitute this value of into the expression the initial direction to be above the horizontal.

• A football quarterback is moving backwards at a cannon that has a muzzle velocity of 4.0 km/s.

• A ship sails from The Netherlands to the range of the super cannon.

• There is a problem in which to calculate the of east.
• How fast the ship is relative to the ball's initial speed to clear the fence.

• The height at which the ball is released.

• Given the distances and heights of the wind on the plane's path, examine the possibility of answers that are in line with your expectations.

• Its direction of travel relative to the air is in the opposite direction of his motion relative to the Earth.
• The to the Earth.

• Observer on shore if the bird flies with the wind.

• The front runner has a speed of 2.20 m/s in a direction east of the Earth, and the second runner has a speed of 3.50 m/s in a direction north of the Earth.
• The wind is 4.20 m/s.

• In the example 3.8, the coin dropped by the airline passenger goes horizontally while falling in the frame of reference of the Earth.

• An ice hockey player is moving at 8.00 m/s when he hits distant galaxies and the puck goes to the goal.
• The puck's speed is proportional to its distance.
• It is 29.0 m/s to the player.
• The line between the center of the Earth and the center of the universe appears to be seen by an observer on the planet.
• The player's frame in the center of the Milky Way has the puck'svelocity making a difference to him.

• Observers will see themselves at the center of the expanding universe, and they will be aware of relative velocities, because it is not possible to locate the center of expansion with the given information.

• An ice hockey player moving across the rink has to shoot the puck backwards in order to get to the goal.

• The five galaxies are above the surface of the Earth.
• The relative velocities between the supplies and the astronauts are related to the distances.

• The two parts of the problem give you some idea of the wind's speed.

• An athlete crosses a 25-m wide river by swimming to a runway in a cross wind.
• You can calculate the angle of the airplane relative to the water by constructing a parallel to the water current problem.
• He needs to reach the opposite side at a must fly relative to the air mass in order to reach his starting point.

• The pilot might have to perform maneuvers west of the plane in order for it to land with its wheels pointing north.