Physics I (PHY111): Motion - KSIU Fall 2024 Lecture Notes PDF

Field of Basic Sciences All Programs Lecture2 : (Physics I (PHY111): Motion) Dr. Shehab E. Ali Date : 14/ 10 / 2024 1 Motion=Kinematics Describes motion while ignoring the external agents that might have ca...

Field of Basic Sciences All Programs Lecture2 : (Physics I (PHY111): Motion) Dr. Shehab E. Ali Date : 14/ 10 / 2024 1 Motion=Kinematics Describes motion while ignoring the external agents that might have caused or modified the motion For now, will consider motion in one dimension Along a straight line Motion represents a continual change in an object’s position. Introduction 2 2 Types of Motion Translational An example is a car traveling on a highway. Rotational An example is the Earth’s spin on its axis. Vibrational An example is the back-and-forth movement of a pendulum. Introduction 4 4  Motion in one dimension Motion in two dimension Motion in three dimension  Chapter 2 Motion in One Dimension 1.Position, Velocity, and Speed 2.Instantaneous Velocity and Speed 3.Analysis Model: Particle Under Constant Velocity 4.Acceleration 5.Motion Diagrams 6.Analysis Model: Particle Under Constant Acceleration 7.Freely Falling Objects 2.1 Position, Velocity, and Speed A particle’s position x is the location of the particle with respect to a chosen reference point that we can consider to be the origin of a coordinate system. The motion of a particle is completely known if the particle’s position in space is known at all times. Consider a car moving back and forth along the x axis as in Figure 2.1a. When we begin collecting position data, the car is 30 m to Pictorial representation the right of the reference position x = 0. We will use the particle model by identifying some point on the car, perhaps the front door handle, as a particle representing the entire car. We start our clock, and once every 10 s we note the car’s position. As you can see from Table 2.1 pictorial representation Tabular representation Graphical representation  Given the data in Table 2.1, we can easily determine the change in position of the car for various time intervals. The displacement Δx of a particle is defined as its change in position in some time interval. As the particle moves from an initial position to a final position , its displacement is given by we see that Δx is positive if is greater than and negative if is less than. It is very important to recognize the difference between displacement and distance traveled. Distance is the length of a path followed by a particle. Distance is always represented as a positive number, whereas displacement can be either positive or negative. Displacement is an example of a vector quantity. Many other physical quantities, including position, velocity, and acceleration, also are vectors. In general, a vector quantity requires the specification of both direction and magnitude. By contrast, a scalar quantity has a numerical value and no direction. Distance is a scalar quantity. It follows that any object always moving to the right undergoes a positive displacement Δx < 0, and any object moving to the left undergoes a negative displacement so that Δx > 0.  The average velocity of a particle is defined as the particle’s displacement Δx divided by the time interval Δt during which that displacement occurs: where the subscript x indicates motion along the x axis. From this definition we see that average velocity has dimensions of length divided by time (L/T), or meters per second in SI units. The average velocity of a particle moving in one dimension can be positive or negative, depending on the sign of the displacement. The average speed of a particle, a scalar quantity, is defined as the total distance d traveled divided by the total time interval required to travel that distance: The SI unit of average speed is the same as the unit of average velocity: meters per second. Unlike average velocity, however, average speed has no direction and is always expressed as a positive number. Notice the clear distinction between the definitions of average velocity and average speed: average velocity (Eq. 2.2) is the displacement divided by the time interval, whereas average speed (Eq. 2.3) is the distance divided by the time interval. Example 2.1 Calculating the Average Velocity and Speed Find the displacement, average velocity, and average speed of the car in Figure 2.1a between positions A and F. Solution Use Equation 2.1 to find the displacement of the car: Use Equation 2.2 to find the car’s average velocity: Use Equation 2.3 to find the car’s average speed: 2.2 Instantaneous Velocity and Speed the instantaneous velocity equals the limiting value of the ratio Δx/Δt as Δt approaches zero: In calculus notation, this limit is called the derivative of x with respect to t, written dx/dt: The instantaneous velocity can be positive, negative, or zero. The instantaneous speed of a particle is defined as the magnitude of its instantaneous velocity. As with average speed, instantaneous speed has no direction associated with it. For example, if one particle has an instantaneous velocity of -25 m/s along a given line and another particle has an instantaneous velocity of +25 m/s along the same line, both have a speed of 25 m/s. Example 2.2 Average and Instantaneous Velocity A particle moves along the x axis. Its position varies with time according to the expression where x is in meters and t is in seconds. The position–time graph for this motion is shown in Figure 2.4a. Because the position of the particle is given by a mathematical function, the motion of the particle is completely known, unlike that of the car in Figure 2.1. Notice that the particle moves in the negative x direction for the first second of motion, is momentarily at rest at the moment t = 1 s, and moves in the positive x direction at times t > 1 s. (A) Determine the displacement of the particle in the time intervals t = 0 to t = 1 s and t = 1 s to t = 3 s.  Solution (B) Calculate the average velocity during these two time intervals. Solution  (C) Find the instantaneous velocity of the particle at t = 2.5 s. Solution 2.3 Analysis Model: Particle Under Constant Velocity If the velocity of a particle is constant, its instantaneous velocity at any instant during a time interval is the same as the average velocity over the interval. That is, Therefore, Equation 2.2 gives us an equation to be used in the mathematical representation of this situation: This equation tells us that the position of the particle is given by the sum of its original position at time t = 0 plus the displacement that occurs during the time interval Δ t. In practice, we usually choose the time at the beginning of the interval to be and the time at the end of the interval to be so our equation becomes Equations 2.6 and 2.7 are the primary equations used in the model of a particle under constant velocity. Whenever you have identified the analysis model in a problem to be the particle under constant velocity, you can immediately turn to these equations. Example 2.4 Modeling a Runner as a Particle A kinesiologist is studying the biomechanics of the human body. (Kinesiology is the study of the movement of the human body. Notice the connection to the word kinematics.) She determines the velocity of an experimental subject while he runs along a straight line at a constant rate. The kinesiologist starts the stopwatch at the moment the runner passes a given point and stops it after the runner has passed another point 20 m away. The time interval indicated on the stopwatch is 4.0 s. (A) What is the runner’s velocity? Solution Having identified the model, we can use Equation 2.6 to find the constant velocity of the runner: (B) If the runner continues his motion after the stopwatch is stopped, what is his position after 10 s have passed? Solution Use Equation 2.7 and the velocity found in part (A) to find the position of the particle at time t = 10 s: 2.4 Acceleration When the velocity of a particle changes with time, the particle is said to be accelerating. For example, the magnitude of a car’s velocity increases when you step on the gas and decreases when you apply the brakes. Let us see how to quantify acceleration. In some situations, the value of the average acceleration may be different over different time intervals. It is therefore useful to define the instantaneous acceleration as the limit of the average acceleration as Δt approaches zero. Example 2.5 Average and Instantaneous Acceleration Solution (B) Determine the acceleration at t = 2.0 s. Solution  2.6 Analysis Model: Particle Under Constant Acceleration Example 2.6 Carrier Landing A jet lands on an aircraft carrier at a speed of 140 mi/h (= 63 m/s). (A) What is its acceleration (assumed constant) if it stops in 2.0 s due to an arresting cable that snags the jet and brings it to a stop? Solution Equation 2.13 is the only equation in the particle under constant acceleration model that does not involve position, so we use it to find the acceleration of the jet, modeled as a particle: (B) If the jet touches down at position xi = 0, what is its final position? Use Equation 2.15 to solve for the final position: 2.7 Freely Falling Objects It is well known that, in the absence of air resistance, all objects dropped near the Earth’s surface fall toward the Earth with the same constant acceleration under the influence of the Earth’s gravity. If we neglect air resistance and assume the free-fall acceleration does not vary with altitude over short vertical distances, the motion of a freely falling object moving vertically is equivalent to the motion of a particle under constant acceleration in one dimension. Therefore, the equations developed in Section 2.6 for the particle under constant acceleration model can be applied. The only modification for freely falling objects that we need to make in these equations is to note that the motion is in the vertical direction (the y direction) rather than in the horizontal direction (x) and that the acceleration is downward and has a magnitude of Therefore, we choose where the negative sign means that the acceleration of a freely falling object is downward.  Example 2.9 A stone thrown from the top of a building is given an initial velocity of 20.0 m/s straight upward. The stone is launched 50.0 m above the ground, and the stone just misses the edge of the roof on its way down as shown in Figure 2.5. (A) Using as the time the stone leaves the thrower’s hand at position A, determine the time at which the stone reaches its maximum height. Solution Use Equation 2.13 to calculate the time at which the stone reaches its maximum height: Substitute numerical values:  (B) Find the maximum height of the stone. Solution As in part (A), choose the initial and final points at the beginning and the end of the upward flight. Set and substitute the time from part (A) into Equation 2.16 to find the maximum height: (C) Determine the velocity of the stone when it returns to the height from which it was thrown. Solution Choose the initial point where the stone is launched and the final point when it passes this position coming down. Substitute known values into Equation 2.17: When taking the square root, we could choose either a positive or a negative root. We choose the negative root because we know that the stone is moving downward at point C. The velocity of the stone when it arrives back at its original height is equal in magnitude to its initial velocity but is opposite in direction. (D) Find the velocity and position of the stone at t = 5.00 s. Solution Choose the initial point just after the throw and the final point 5.00 s later. Calculate the velocity at D from Equation 2.13: Use Equation 2.16 to find the position of the stone at What if the throw were from 30.0 m above the ground instead of 50.0 m? Which answers in parts (A) to (D) would change? Answer None of the answers would change. All the motion takes place in the air during the first 5.00 s. (Notice that even for a throw from 30.0 m, the stone is above the ground at t = 5.00 s.) Therefore, the height of the throw is not an issue. Mathematically, if we look back over our calculations, we see that we never entered the height of the throw into any equation. 27