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Chapter 17: Average and Instantaneous Motion

- The distance and displacement are the same quantity.

- We are dealing with a quantity if we consider the actual distance traveled by the object along a path.

- Since directions are specified, this is a number.
- The average speed is the ratio of the total distance traveled to the total elapsed time.
- The units of speed are meters per second.

- If we want to know the instantaneous velocity at any instant in time, we can define it as v, as determined at any precise instant in time.
- If we take into account the direction of motion, instantaneous speed can become velocity.

- Speed is a quantity and velocity is a quantity.

- The graph might look like this if we plotted velocity versus time for this motion.

- For any time interval, the area under the graph equals the displacement.

- If either direction or speed changes, the vechicle will change.
- We say that the object is speeding up if the velocity is changing.
- The object has uniform acceleration if the velocity is changing uniformly.
- A graph of velocity versus time would look like that.
- The slope of the graph is zero.
- If the velocity is constant, acceleration is zero.

- The rate of change of velocity is equal to the quantity of acceleration.

- The area of the triangle is equal to the displacement from t to any other time.
- Between two times, the figure is a trapezoid.
- The displacement versus time graph for uniformly accelerated motion is a parabola if we make the initial conditions that are t, x, and v.

- The average acceleration in units of meters per second squared is known as the slope of the velocity versus time graph.

- An object with zero acceleration is not at rest in a frame of reference.
- The object may be moving quickly.

- If the acceleration is constant in time, our expression for average acceleration can be written in a way that allows us to calculate the instantaneous final velocity after a period of acceleration has taken place.

- We get a graph that looks like this if we plot velocity versus time for uniformly accelerated motion.

- The total area under the graph will be equal to the displacement during any period of time.
- The total area will be the sum of two areas, one a triangle and the other a rectangle.
- The area of the rectangle has been just vit for some time.
- The area of the triangle is half the height.
- The time period is the base and the height is the change in velocity.
- The area of the triangle is 1/2 D vt.

- An alternative method of determining the average speed of an object is suggested by this analysis.
- The area halfway between vi and vf will be the same.

- Sometimes a problem in kinematics doesn't mention the time involved.
- It would be nice to have a formula that does not involve the time factor.
- All the other formulas can be used to derive one.

- Since vf + vi, we can express the time as t.

- The motion can be seen graphically.
- Each region of the graph has a negative, positive, or zero average acceleration.

- The instantaneous slopes on the time graph are the same as the instantaneous slopes on the position.
- We need to look at how the slopes are changing over the interval.

- All instantaneous slopes have the same value.
- There is no change in speed.

- The instantaneous slopes are decreasing from large positive to zero during this interval.

- Positive acceleration is increasing from small positive values to large ones.

- We need to consider the convention accepted for dealing with different directions since velocity and acceleration are vector quantities.
- We usually agree to consider the motion up or down as positive or negative.
- Negative velocity means motion to the left and does not mean the object is slowing down.
- The object is not slowing down if negative acceleration is used.
- The object is slowing down when people say it is decelerating.
- It can happen in two different ways.
- An object with a positive speed can slow down.
- An object with a negative speed can slow down.
- The general rule is that if the two are pointing in the same direction, the object will speed up.
- The object slows down if they are in opposite directions.

- The direction of the acceleration can be found from doing a quick head-to-tail graphical solution of v f - v i.

- The downward acceleration is provided by gravity.
- The symbol g is used to represent the value of gravity on Earth.
- An object going upward will be slowed down by gravity, while a downward moving object will be sped up.
- Even if the object is moving vertically up or down, this is still true.
- If the object is moving sideways under the influence of gravity, it's called free fall.

- During free fall, the acceleration vector never changes.

- If an object falls, vi is 0.
- All subsequent displacements and velocities are negative.
- vi is positive if an object is thrown upward.
- vi is negative if an object is thrown downward.

- A projectile is fired at a high rate of speed.

- The graphical analysis of motion can give us a lot of information.
- If complex changes in motion are taking place, visualization may provide a better understanding of the physics involved.
- The techniques of graphical analysis are easy to use.
- For uniformly accelerated motion, the graph of distance versus time is a parabola.
- The instantaneous velocity can be approximated by finding the slope of a line drawn to a point on the curve.

- We can use graphs of velocity and acceleration in displacement versus time to represent the motion.

- Many instances of motion can be applied to graphs.
- The graph of velocity versus time for an object thrown upward into the air, reaching its highest point, changing direction, and then slowing down is used in the case of changing velocity.
- At first, this motion slows the object down, but later it speeds it up.
- There is a graph of this motion.

- A ball is thrown into the air and caught by a person 5 m above the ground.

- Since the ball is thrown upward, the initial velocity is positive and the acceleration of gravity is always downward.
- We can use the above equation to find the initial velocity when y is 5 m and t is 9 s.

- The ball's speed becomes zero when it rises to its maximum height.

- The maximum height is equal to the value of y when the final velocity is zero.
- We find that y max is 101.7 m using our answer and the known value of g.

- 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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