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Chapter 21: Relative Motion

- All measurements are relative.
- All measurements, including velocity, are made with reference to an object.
- All velocities are relative to a specific coordinate system.
- What we mean when we say a car is moving at 55 mph is 55 mph relative to Earth's surface.
- The means by which to translate one relative velocity to another are provided by the rules of vector addition.

- An example of this type of motion can be seen when a boat tries to cross a river or an airplane tries to cross a crosswind.
- The boat's speed is determined by the properties of the engine and is measured by the speedometer on board.
- The relative velocity is different for a person on the shore than it is for a boat.
- The river is moving to the right at 4 meters per second and the boat is moving at 10 meters per second.

- The result is given by the Pythagorean theorem.
- The direction is found by connecting the head to the tail with a simple sketch.
- We can use it numerically.

- At an angle of 22 degrees east of north, the velocity is 10.5 meters per second.

- If you roll a ball off a table, you will see that it doesn't fall straight down.
- You can observe how far it will fall with trial and error.
- The ball has no vertical speed.
- The ability to "fall" is determined by gravity, and it takes 9.8 meters per second to fall.
- The two motions are independent since gravity acts vertically and the initial velocity is horizontal.
- The trajectory characterized by constant horizontal velocity and constant vertical acceleration is called a parabola by Galileo.

- Since the time is the same for both motions, we can first solve for the time using the x equation and then substitute it for the y equation.

- The equation of y in terms of x is called the trajectory of the projectile, while the two separate equations for x and y as functions of time are called parametric equations.

- The two component motions are independent of each other.
- The horizontalvelocity is constant throughout the projectile's motion.

- The time to fall can be calculated from the equation if the height of the projectile is known.
- The time to fall is 3.16 s if the height is 49 m.

- A projectile is launched from a height of 25 m and is seen to land 50 m from the base.

- The vertical motion is independent of the horizontal motion.

If a rocket is launched with some initial speed at an angle, what would it look like?

- Since each motion is independent, we can consider the fact that the horizontal velocity will be constant while the y velocity will decrease as the rocket rises.
- The rocket's vertical velocity will be zero when it reaches its maximum height.
- At a constant rate, it will move forward.

- This time, the total time of flight will be twice as long.
- The range is the result of the initial horizontal velocity and total time.

- At the maximum height, the vertical velocity component is equal to zero.
- When launching and landing at the same height, the maximum range occurs when the launch angle is equal to 45deg.

- If we want to know the maximum height achieved, we just use the time to reach the highest point.
- A baseball is being hit.

- The maximum range will be equal to 883.7 m if a projectile is launched with an initial velocity of 100 m/s at an angle of 30deg.

- The maximum height reached is 127.55 m.

- We know that viy is v 0 sin th.

- Avector is the name of the thing.
- A change in direction is a change in the vector, so an acceleration is required to change a velocity's direction.
- The direction of the velocity will be changed by the direction of the acceleration.
- Uniform circular motion is achieved when the direction is the only quantity changing.
- An object is undergoing periodic, uniform circular motion.
- We mean that the object maintains a constant speed as it revolves around a circle for a period of time.
- The number of revolutions per second is called the Frequency.

- The centripetal acceleration is the direction of the acceleration.

- It is more convenient to describe the motion in terms of the radians.
- The rate of change of the position of the body is also known as the rate of rotation.

- The mass is moving with a constant speed of 10 m/s in a circle of 2 m.

- All three of the displacement, velocity, and acceleration are related.

- The distance, speed, and time are all variables.

- There is a description of motion.

- The rate of change of displacement is equal to the quotient.

- The rate of change of velocity is what Acceleration is defined to be.

- The slope of a displacement versus time graph is called the vechicle.

- The slope of a graph is called the Acceleration.

- The area under the graph can be used to get the displacement.

- The acceleration is caused by gravity near the surface of Earth and is directed downward.
- This is the only way to get rid of free fall and projectile motion problems.

- The relative velocity can be found by adding the individual velocities.

- In the absence of air resistance, the horizontal motion is independent of the vertical motion.

- The launch velocity can be obtained using the vertical and horizontal components.
- The regular equations can be used for each direction.

- The centripetal acceleration is when an object moves in a circle.

- When you solve a physics problem, be sure to consider the assumptions being made about the moving object.
- You will be able to keep track of what is relevant for your solution path in this way.

- The goals may be explicit or implicit.
- If a question is based on a decision or prediction, you need to understand the requirements to reach an answer.

- Consider the meaning of your solution.

- Remember the sign conventions for treating vector quantities when choosing a coordinate system.
- Make sure you understand the nature of the concepts being discussed.

- Make sure that correct units are included in your final answer by using proper SI units throughout your calculations.
- Try to figure out if the answer makes sense.
- Maybe it looks too large or small because it's expressed in the wrong units.

- If no sketch is provided, make one.

- If you are interpreting a graph, you need to understand the interrelationships of all the variables.

- If you want to make a graph, you need to label both axes, choose a scale for each axis, and draw clearly.

- If you get stuck on a difficult problem, try different problem-solving tricks.

- If the problem is two-dimensional, break the vectors into components first.

- A ball with an initial speed of 20 m/s is thrown upward.

- A plane lands on a runway with a speed of 150 m/s.

- A ball is thrown from a roof at 25 m/s.

- After 25 m/s 2 for 5 s, the engine is shut off and the rocket continues to move upward.

- The velocity versus time graph is shown below.

- An object has a speed of 15 m/s.

- A projectile is launched with a speed of 250 m/s.

- A projectile is launched from the top of a 75-m height.

- A projectile is launched.
- 500 m away, it hits the top of a building.

- The operator of a boat wants to cross a 5-km wide river that is flowing to the east at 10 m/s.
- He wants to reach the exact point on the opposite shore after 15 minutes.

- The graph of velocity versus time can be made if the particle begins its motion at t.

- A stone is dropped.
- A second stone is thrown downward at the same time that the first stone hits the ground.

- A particle is moving in one direction.

- A stone is dropped from a height and falls in 4 seconds.

- A girl standing on top of a roof throws a stone into the air.
- She throws a stone with the same speed.
- When the stones reach the ground, compare their velocities.

- A mass attached to a string is twirled overhead in a horizontal circle.
- The mass lands 2.6 m away.

- A football quarterback throws a pass to a receiver at an angle of 25 degrees to the horizontal and at an initial speed of 25 m/s.
- The quarterback is 30 m from the receiver.
- The receiver runs at a constant pace to catch the ball.

- A car is moving in a straight line.
- The raindrops are falling with a constant terminal velocity.

- We need to know how long it will take to decelerate the ball.
- The answer is 2.04 s if you divide 20 m/s by 9.8 m/s 2.

- The rocket has a speed of 125 m/s after 5 s of acceleration.
- The rocket is decelerated by gravity as it continues to move upward after the engine stops.
- The time to decelerate to zero is found by dividing 125 m/s by gravity.

- The first accelerated distance is added to the distance traveled during that time.

- The total distance traveled is equal to 1,100 m.

- The areas of the triangles and rectangles add up to 62.5 m.

- 35 - 2.5 - 20 - 5 is the final displacement.

- The change in displacement from question 6 is 7.5 m in 14 s and the average velocity is 0.53 m/s.

- If the average speed is 30 m/s, the final speed must be 45 m/s.
- A change of 30 m/s at a rate of 3 m/s 2 for 10 s is implied.

- We get vx with the known numbers, which is 175 m/s.

- We get 3.81 s for the time if we substitute known numbers.

- Therefore, vi cos th is 125, vi sin is 32.1, and tan is 0.2568.

- The Pythagorean theorem states that the boat must be at least 11 m/s.
- The angle is given by the function.
- The W of N is 60.9deg.

- The graph shows the constant acceleration from t to t. The area has a constant change in speed from 0 to 9 m/s.
- The object slows down and turns around in the second region.
- The area is -12 m/s.
- The graph went from 9 m/s to 3 m/s.
- The last area has a change of 3 m/s.

- The first stone will be dropped from a height of 75 m, and it will be in free fall because of gravity.

- Since the first stone is to fall 15 m before the second, we can determine that t is 1.75 s to fall that distance.
- The second stone must reach the ground after the first one has traveled for 3.8 s.

- The second stone has a downward initial velocity of 24.34 m/s.

- The average velocity is equal to 15 m/s for D x and D t. The new average velocity is equal to 13.5 m/s.
- We are continuing this procedure.
- The average is 12.6 m/s.
- The average is 12.3 m/s.
- The average is 12.03 m/s.
- The average velocity is 12.003 m/s.

- The instantaneous velocity is equal to 12 m/s at 2 s.

- T is the total time to fall.

- The stone falls in 4 seconds.

- h is the number of meters, and T is the number of seconds.

- The average velocity is the ratio of the change in displacement to the change in time.
- It is possible that the object stops and continues.
- It is possible to have a zero instantaneous velocity.

- When the stones reach the ground, they will have the same speed.
- When the first stone rises, gravity decelerates it until it stops, and then it falls back down.
- It has the same speed but in a different direction when it passes its starting point.
- The starting speed is the same as the second stone.
- Both stones are accelerated through the same displacement, giving them the same final velocities.

- The total displacement is divided by the total time to arrive at the average velocity.
- The average speed is the same as the total distance divided by the time.
- An object can have zero displacement in one period if it returns to its starting point.
- It has zero average velocity.
- Since it traveled a long way, it has an average speed.

- The mass is moving in a straight line.
- It has a constant speed.
- The height of the mass and the horizontal range can be used to determine thevelocity when it is released.

- We find that the r is 0.288 m.

- The receiver has to travel 18.85 m away from the quarterback to catch the ball.
- To determine how fast the receiver must run, we need to know how long it takes the ball to travel.

- To travel 18.85 m in 2.16 s, the receiver needs to run at 8.73 m/s.

- The driver can see that from the diagram below.

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