1/27. So we get for your average speed: The average velocity uses the same time, since the time in both cases is just the time for the

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1 1/27 Example: A Trip Home - Suppose you are visiting a friend. It is time to leave and you remember that you have to pick up a quart of milk on the way home. Your route is shown in the figure below. We want to calculate your average speed on the way home and your average velocity on the way home. First, let s calculate your average speed. We need to know the total distance traveled, which is just the sum of the distance from your friend s house to the store and the distance from the store to your home: W e al so need to find the time for the trip, which is the sum of the time to travel from your friend s house to the store, the time to buy the milk, and the time to travel from the store to your home: This is a little harder because we don t know times for the traveling parts of the trip. However, we know that the time to travel a certain distance at a certain speed is just the ratio of the distance to the speed, so we have So we get for your average speed: The average velocity uses the same time, since the time in both cases is just the time for the

2 whole trip. However, as we know, the numerator is the change in coordinate. Your final coordinate is the coordinate of your home and your initial coordinate is the coordinate of your friend s house. So, the average velocity is We see that they are substantially different and have opposite signs! Instantaneous Velocity and Speed The instantaneous speed of a particle is the speed that it has at a single instant of time. The speedometer in your car measures your instantaneous speed. At the instant the needle on your speedometer crosses the 35 mph line, you are traveling 35 mph. Instantaneous velocity is the velocity at an instant of time. Note that, in one dimension, instantaneous speed and velocity always have the same numerical size. This is due to the fact that, at an instant of time, it is not possible for a particle to travel past a point and then back. The difference is direction. Speed is always positive, but velocity is positive for motion to the right and negative for motion to the left (given a standard number line as a coordinate system). In one dimension, instantaneous speed is the absolute value of the instantaneous velocity. Speed is how fast. Velocity is how fast and in what direction. What happens if the velocity changes? Acceleration! Acceleration Acceleration is the rate at which velocity changes. Acceleration can involve a change in speed, a change in direction, or both. Note that a decreasing velocity is also an acceleration. A deceleration is an acceleration that is opposite the velocity. Note that the speed can be constant and a body can still accelerate because the direction of its motion changes moving round in a circle with constant speed. Since acceleration is defined as the rate at which velocity changes, the average acceleration is given by the change in velocity divided by the time taken for the change to take place. In symbols,

3 Galileo determined that the acceleration of a falling body is constant because such a body obeys the expected formula. He did this by rolling balls down an inclined plane and measuring the time for the ball to travel various distances. Let s see if we can work out this formula. We will work with speed and distance. We need to know two things. We will assume that we drop a body from rest. The distance it travels after falling for a given amount of time is Distance = avg speed time or This speed is the average speed of the body over the time interval t. The speed of a body after traveling for a given amount of time is at a given acceleration is speed = acc time or Note that this speed is NOT the average speed it is the speed of the body after falling for time t. We see that speed increases at a uniform rate, so avg speed is the avg of the initial and final speeds for a time interval. Since the initial speed is zero and the final speed is v, the average speed is just average speed = ½ final speed or Combining this expression for the average speed with the one for distance above, we get distance = ½ speed time =1/2 acc time time = ½ acc time 2 or Note: the unit of acceleration is m/s/s = m/s 2. Let s work this out for a falling body. A falling body has acceleration g = 9.8 m/s 2, which is roughly 10 m/s 2. We will use the latter number because it is easier to work with. We will make a table. The first column will be the time in seconds. The second column will be the speed of fall at that time. The third column will be the average speed over that time interval. The fourth column will be the distance calculated from the product of the average speed and the time, and the fifth column, the distance found from the formula we derived.

4 time (s) t speed (m/s) v = at avg speed (m/s) v = ½v dist (m) d = vt dist (m) The distance of fall as a function of time is shown by the plus signs below (horizontal because there isn t enough room to place it vertically. 0 1 s 2 s 3 s 4 s 5 s m 20 m 45 m 80 m 125 m Chapter 4 Newton s Second Law of Motion Newton s Second Law of Motion (N2): If a body suffers an external influence, we say it has a net force applied to it. A net force on a body causes its velocity to change, that is, it causes the body to accelerate. The net force on the body is proportional to the acceleration of the body with the constant of proportionality being defined to be the mass of the body. Newton s second law does four things: 1. Operationally defines force (gives a way that, in principle, force can be measured) 2. Operationally defines mass. 3. Given the net force on a body, it allows us to calculate the motion of the body. 4. Given the desired trajectory of a body, it allows us to calculate the force needed to produce the path. Operational definition of force. Use a spring to apply a force to a standard (1 kilogram = 1kg) mass. Cause the mass to accelerate at 1 m/s 2 and note the length of the spring. Repeat for 2, 3, 4, 5,

5 etc, m/s 2 to calibrate the spring. Once the spring is calibrated, it can be used to measure forces and to apply known forces to other systems. Unit of force: a newton = N is the force that causes a 1 kg mass to accelerate at 1 m/s 2. Operational definition of mass. Use the calibrated spring to apply a 1-N force to a block of mass m. Measure the acceleration of the block, and use N2 (Newton s second law) to solve for the mass. Note about mass: If the 1-N force can only produce a small acceleration, the mass is large. If the 1-N force produces a large acceleration, the mass is small. Thus, mass is a measure of how hard it is to change the motion of a body; it is a measure of the body s inertia. Note that the mass of a body is independent of the direction it is pushed it is the same back and forth, front and back, up and down. Note that the mass of body is a characteristic of a body and does not change with its environment. We want to talk about the difference between weight and mass. Weight We will define the weight of a body as the force of gravity on the body. The weight of a body is the total force on a body in free fall. This force must be equal to the product of its mass and its acceleration, which is the acceleration due to gravity g. Note that the weight of a body is given by this expression regardless of the body s acceleration. A body supported has no net force on it weight is balanced by the support force. The term weightless is a misnomer. The term is often used to describe conditions in spacecraft in space. But the astronauts are NOT weightless the still experience the force of gravity of the Earth on them. The term that should be used is free fall. The distinction between weight and mass. 1. Weight is a force. It has a direction and is a vector. Mass is a scalar it has no direction associated with it. 2. Weight is the force of gravity on a body; mass is a measure of the body s inertia.

6 3. Mass is independent of location a 1-kg mass has a mass of 1 kg anywhere in the universe. Weight depends on where the body is located because the acceleration due to gravity is different in different locations. Chapter 5 Newton s Third Law of Motion Most often heard stated as To every action, there is an equal and opposite reaction. Open to misinterpretation. Great care must be taken in applying scientific principles outside the discipline for which originally designed testing and experimentation scientific method. Best form of N3 not open to misinterpretation: Newton s Third Law (N3): If two bodies A and B interact, the force that A applies to B is equal in size but opposite in direction to the force that B applies to A. Note these forces are always equal and opposite no matter what the motion of the two objects. Push a stool. The stool moves because the push is greater than the friction on the stool. The pusher doesn t move because the equal and opposite push of stool on pusher is not greater than the friction on the pusher. On a frictionless surface each would move away from the other. If on a frictionless surface, the pusher pushes the stool, the pusher ends up with less speed because the pusher s mass is greater, which means smaller acceleration than the stool. Note that N3 is only precisely true for contact forces good approximation otherwise. For action at a distance, time is required for a change in the condition of one body to be reflected in the force on a second body due to the finite speed of propagation. Use the field concept to obtain same results as N3 one body creates a field in its vicinity and this field applies the force to the second body. We will always use N3. For Homework: Type the question. Separate answer from question space or bold or etc. Provide explanation of your reasoning.

7 If a physical concept is involved, always mention the concept. For instance, for the ball rolling and coming to rest question: Newton s first law tells us that a body in motion will continue in motion with constant speed unless acted upon by an external force. Since the body is not traveling with constant speed, there must be a force. This force is friction.

Chapter 6 Dynamics I: Motion Along a Line

Chapter 6 Dynamics I: Motion Along a Line Chapter Goal: To learn how to solve linear force-and-motion problems. Slide 6-2 Chapter 6 Preview Slide 6-3 Chapter 6 Preview Slide 6-4 Chapter 6 Preview Slide