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**Graphing Ideas in Physics And Use of Vectors**

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1.4 Simple Types of Motion Below is a table of values of time and distance. The table shows the values of distance (d) at certain times (t). These values all satisfy the equation d = 7t. You could make the table longer or shorter by using different time increments.

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1.4 Simple Types of Motion A way to show the relationship between distance and time is to graph the values in the table. The usual practice is to graph distance versus time, which puts distance on the vertical axis. Note that the data points lie on a straight line.

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1.4 Simple Types of Motion Remember this simple rule: when the speed is constant, the graph of distance versus time is a straight line. In general, when one quantity is proportional to another, the graph of the two quantities is a straight line.

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1.4 Simple Types of Motion An important feature of this graph is its slope. The slope of a graph is a measure of its steepness. In particular, the slope is equal to the rise between two points on the line divided by the run between the points.

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**1.4 Simple Types of Motion This is illustrated in the figure shown.**

The rise is a distance, Dd, and the run is a time interval, Dt. So the slope equals Dd divided by Dt, which is also the object’s velocity. The slope of a distance- versus-time graph equals the velocity.

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1.4 Simple Types of Motion The graph for a faster-moving body, a racehorse for instance, would be steeper—it would have a larger slope. The graph of d versus t for a slower object (a person walking) would have a smaller slope. When an object is standing still (when it has no motion), the graph of d versus t is a flat line parallel with the horizontal axis. The slope is zero because the velocity is zero.

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1.4 Simple Types of Motion Even when the velocity is not constant, the slope of a d versus t graph is still equal to the velocity. In this case, the graph is not a straight line because, as the slope changes (a result of the changing velocity), the graph curves or bends.

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1.4 Simple Types of Motion The graph shown represents the motion of a car that starts from a stop sign, drives down a street, and then stops and backs into a parking place.

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**1.4 Simple Types of Motion When the car is stopped, the graph is flat.**

The distance is not changing and the velocity is zero. When the car is backing up, the graph is slanted downward. The distance is decreasing, and the velocity is negative.

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**1.4 Simple Types of Motion Constant Acceleration**

Keep in mind that the graph of distance versus time is a straight line for uniform motion. For constant acceleration it is the graph of velocity versus time that exhibits a linear behavior.

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**1.4 Simple Types of Motion Constant Acceleration**

As shown in the table of distance values for a falling body, distance increases rapidly.

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**1.4 Simple Types of Motion Constant Acceleration**

The graph of distance versus time curves upward. This is because the velocity of the body is increasing with time, and the slope of this graph equals the velocity.

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1.4 Simple Types of Motion Rarely does the acceleration of an object stay constant for long. As a falling body picks up speed, air resistance causes its acceleration to decrease. When a car is accelerated from a stop, its acceleration usually decreases, particularly when the transmission is shifted into a higher gear.

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1.4 Simple Types of Motion The figure shows the velocity of a car as it accelerates from 0 to 80 mph. Note that acceleration steadily decreases (the slope gets smaller). During the short time the transmission is shifted, the acceleration is zero.

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1.4 Simple Types of Motion During a karate demonstration, a concrete block is broken by a person’s fist.

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1.4 Simple Types of Motion The fist travels downward until it contacts the block, at about 6 milliseconds. This causes a large acceleration as the fist is brought to a sudden stop.

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1.4 Simple Types of Motion The graph shows the velocity of the fist, just the slope of the distance versus time graph at each time. Contact with the concrete is indicated by the steep part of the graph as the velocity goes to zero.

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1.4 Simple Types of Motion If we take the slope of this segment of the velocity graph, we find that the magnitude of the acceleration of the fist at that moment is about 3,500 m/s2, or 360g (ouch!). What happened at about 25 milliseconds?

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**2-8 Graphical Analysis of Linear Motion**

This is a graph of x vs. t for an object moving with constant velocity. The velocity is the slope of the x-t curve. © 2014 Pearson Education, Inc.

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**2-3 Instantaneous Velocity**

On a graph of a particle’s position vs. time, the instantaneous velocity is the tangent to the curve at any point. Figure Caption: Graph of a particle’s position x vs. time .The slope of the straight line P1P2 represents the average velocity of the particle during the time interval Δt = t2 – t1. Figure Caption: Same position vs. time curve as in Fig. 2–10, but note that the average velocity over the time interval ti – t1 (which is the slope of P1Pi) is less than the average velocity over the time interval t2 – t1. The slope of the thin line tangent to the curve at point P1 equals the instantaneous velocity at time t1.

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Concept Map 1.2

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**3-1 Vectors and Scalars A vector has magnitude as well as direction.**

Some vector quantities: displacement, velocity, force, momentum A scalar has only a magnitude. Some scalar quantities: mass, time, temperature

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**Quantities that have both a magnitude and a direction.**

Vectors Quantities that have both a magnitude and a direction. Magnitude consists of an amount and the units. Direction can be expressed as some defined "x-hat" or "y-hat" direction or as East, North, etc. Example: A displacement vector (distance with direction): 40 meters East

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**Drawing Vectors We draw a vector as:**

An arrow from tail to head with the head of the arrow pointing in the direction of the vector. The length of the arrow represents the vector's magnitude We specify its direction by the angle it makes with the (+) horizontal axis. head tail

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**3-2 Addition of Vectors—Graphical Methods**

For vectors in one dimension, simple addition and subtraction are all that is needed. You do need to be careful about the signs, as the figure indicates.

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**Adding Vectors To Add a Set of Vectors**

Draw the first vector from a start point on a grid with horizontal and vertical axes. Draw the next vector starting from the head of the previous vector. Repeat this last step until the last vector is drawn. Now draw the resultant (or “net”)vector starting from the tail of the first vector to the head of the last vector (or in other words: from start point to end point).

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**1.2 Speed and Velocity Vector Addition**

Sometimes a moving body has two velocities at the same time. The runner on the deck of the ship in the figure has a velocity relative to the ship and a velocity because the ship itself is moving.

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**1.2 Speed and Velocity Vector Addition**

A bird flying on a windy day has a velocity relative to the air and a velocity because the air carrying the bird is moving relative to the ground. The velocity of the runner relative to the water or that of the bird relative to the ground is found by adding the two velocities together to give the net, or resultant, velocity. Let’s consider how two velocities (or two vectors of any kind) are combined in vector addition.

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**1.2 Speed and Velocity Vector Addition**

When adding two velocities, you represent each as an arrow with its length proportional to the magnitude of the velocity—the speed. For the runner on the ship, the arrow representing the ship’s velocity is a little more than 2 times as long as the arrow representing the runner’s velocity because the two speeds are 20 mph and 8 mph, respectively.

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**1.2 Speed and Velocity Vector Addition**

Each arrow can be moved around for convenience, provided its length and its direction are not altered. Any such change would make it a different vector.

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**1.2 Speed and Velocity Vector Addition**

The procedure for adding two vectors is as follows. Two vectors are added by representing them as arrows and then positioning one arrow so its tip is at the tail of the other. A new arrow drawn from the tail of the first arrow to the tip of the second is the arrow representing the resultant vector—the sum of the two vectors.

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**1.2 Speed and Velocity Vector Addition**

The figure shows this for the runner on the deck of the ship. The runner is running forward in the direction of the ship’s motion, so the two arrows are parallel. When the arrows are positioned “tip to tail,” the resultant velocity vector is parallel to the others, and its magnitude—the speed—is 28 mph (8 mph +20 mph).

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**1.2 Speed and Velocity Vector Addition**

In this figure, the runner is running toward the rear of the ship, so the arrows are in opposite directions. The resultant velocity is again parallel to the ship’s velocity, but its magnitude is 12 mph (20 mph – 8 mph).

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**1.2 Speed and Velocity Vector Addition**

Vector addition is done the same way when the two vectors are not along the same line. The figure shows a bird with velocity 8 m/s north in the air while the air itself has velocity 6 m/s east.

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**1.2 Speed and Velocity Vector Addition**

The bird’s velocity observed by someone on the ground, (b), is the sum of these two velocities.

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**1.2 Speed and Velocity Vector Addition**

We determine this by placing the two arrows representing the velocities tip to tail as before and drawing an arrow from the tail of the first to the tip of the second. The direction of the resultant velocity is toward the northeast.

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**3-2 Addition of Vectors—Graphical Methods**

If the motion is in two dimensions, the situation is somewhat more complicated. Here, the actual travel paths are at right angles to one another; we can find the displacement by using the Pythagorean Theorem.

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**3-2 Addition of Vectors—Graphical Methods**

Adding the vectors in the opposite order gives the same result:

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**1.2 Speed and Velocity Vector Addition**

Watch for this when you see a bird flying on a windy day: Often the direction the bird is moving is not the same as the direction its body is pointed.

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**1.2 Speed and Velocity Vector Addition**

What about the magnitude of the resultant velocity? It is not simply or 8 – 6, because the two velocities are not parallel. With the numbers chosen for this example, the magnitude of the resultant velocity—the bird’s speed—is 10 m/s.

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**1.2 Speed and Velocity Vector Addition**

If you draw the two original arrows with correct relative lengths and then measure the length of the resultant arrow, it will be 5/4 times the length of the arrow representing the 8 m/s vector. Then 8 m/s times 5/4 equals 10 m/s. This can be calculated using the Pythagorean theorem because the arrows form a right triangle.

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**1.2 Speed and Velocity Vector Addition**

Vector addition is performed in the same manner, no matter what the directions of the vectors. The figure shows two other examples of a bird flying with different wind directions. The magnitudes of the resultants in these cases are best determined by measuring the lengths of the arrows.

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**1.2 Speed and Velocity Vector Addition**

There are many other situations in which a body’s net velocity is the sum of two (or more) velocities (for example, a swimmer or boat crossing a river). Displacement vectors are added in the same fashion. If you walk 10 meters south, then 10 meters west, your net displacement is 14.1 meters southwest.

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**1.2 Speed and Velocity Vector Addition**

The process of vector addition can be “turned around.” Any vector can be thought of as the sum of two other vectors, called components of the vector. When we observe the bird’s single velocity, we would likely realize that the bird has two velocities that have been added.

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**1.2 Speed and Velocity Vector Addition**

A soccer player running southeast across a field can be thought of as going south with one velocity and east with another velocity at the same time.

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**3-2 Addition of Vectors—Graphical Methods**

Even if the vectors are not at right angles, they can be added graphically by using the tail-to-tip method.

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**3-2 Addition of Vectors—Graphical Methods**

The parallelogram method may also be used; here again the vectors must be tail-to-tip.

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**3-3 Subtraction of Vectors, and Multiplication of a Vector by a Scalar**

In order to subtract vectors, we define the negative of a vector, which has the same magnitude but points in the opposite direction. Then we add the negative vector.

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**3-3 Subtraction of Vectors, and Multiplication of a Vector by a Scalar**

A vector can be multiplied by a scalar c; the result is a vector c that has the same direction but a magnitude cV. If c is negative, the resultant vector points in the opposite direction.

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**3-4 Adding Vectors by Components**

Any vector can be expressed as the sum of two other vectors, which are called its components. Usually the other vectors are chosen so that they are perpendicular to each other.

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**3-4 Adding Vectors by Components**

If the components are perpendicular, they can be found using trigonometric functions.

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**3-4 Adding Vectors by Components**

The components are effectively one-dimensional, so they can be added arithmetically.

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**3-4 Adding Vectors by Components**

Draw a diagram; add the vectors graphically. Choose x and y axes. Resolve each vector into x and y components. Calculate each component using sines and cosines. Add the components in each direction. To find the length and direction of the vector, use: and .

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**3-4 Adding Vectors by Components**

Example 3-2: Mail carrier’s displacement. A rural mail carrier leaves the post office and drives 22.0 km in a northerly direction. She then drives in a direction 60.0° south of east for 47.0 km. What is her displacement from the post office?

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**3-4 Adding Vectors by Components**

Example 3-3: Three short trips. An airplane trip involves three legs, with two stopovers. The first leg is due east for 620 km; the second leg is southeast (45°) for 440 km; and the third leg is at 53° south of west, for 550 km, as shown. What is the plane’s total displacement?

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**3-5 Unit Vectors Unit vectors have magnitude 1.**

Using unit vectors, any vector can be written in terms of its components:

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3-6 Vector Kinematics In two or three dimensions, the displacement is a vector:

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3-6 Vector Kinematics As Δt and Δr become smaller and smaller, the average velocity approaches the instantaneous velocity.

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3-6 Vector Kinematics The instantaneous acceleration is in the direction of Δ = 2 – 1, and is given by:

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**2-9 Graphical Analysis and Numerical Integration**

The total displacement of an object can be described as the area under the v-t curve:

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