PHYS101: Introduction to Mechanics

2a. Compare and contrast distance and displacement

• Define distance.
• Define displacement.
• Give an example of motion when the distance and displacement are the same.
• Give an example of motion when the distance and displacement are different.

Distance describes how much an object has moved. It depends on how the object has moved, that is, the path the object took to get from the starting point to the ending point. The units for distance are length units, such as meters. Distance is called a scalar quantity. A scalar quantity describes the magnitude of the measurement, but not a specific direction.

Displacement describes the overall change in position of an object. It only depends on the starting and ending point of the object. It does not depend on the path taken to get between the two points. Like distance, the units for displacement are length units, such as meters. Displacement is a vector quantity, which means it has a magnitude and a specific direction associated with the measurement. So, the complete units for displacement also include a direction.

For an example, consider a four-story building. A person needs to travel on the elevator from the first to the third floor. To accomplish this, the person could take an elevator directly from the first floor to the third floor. In this case, the distance and displacement are the same, because the person went directly from the starting to the ending point.

However, this is not the only way the person could travel from the first to the third floor. They could accidentally hit the fourth floor button when they got on the elevator. In this case, they would travel from the first floor to the fourth floor, and back down to the third floor. In this instance, the displacement is still the first floor to the third floor. But, the distance is longer, because the person took a detour to the fourth floor.

2b. Define and distinguish between vector and scalar physical quantities

• What is a scalar physical quantity? Give an example of a scalar physical quantity.
• What is a vector physical quantity? Give an example of a vector physical quantity.
• What is the major difference between scalar and vector physical quantities?

A scalar physical quantity is a measurement of quantity that has a magnitude (amount), but not a direction. Examples of scalar quantities include mass and temperature. These are scalar quantities because there is no direction associated with these measurements. In the previous learning outcome, we saw that distance is a scalar quantity because it has no direction associated with it.

A vector physical quantity is a measurement that has a magnitude (amount) and direction. Vectors are often depicted as an arrow. The length of the arrow shows the magnitude of the quantity, and the direction of the arrow shows the direction of the vector.

For simple one-dimensional systems, a vector is often written as the magnitude with a (+) or (-) to indicate direction. As we saw in the previous learning outcome, displacement is a vector quantity. Velocity is also a vector quantity, for example, 5.5 km/s east. This measurement shows the magnitude of the velocity (5.5 km/s) and the direction (east).

The major difference between scalar and vector quantities is that scalar quantities only have a magnitude and vector quantities have a magnitude and direction.

Review Vectors, Scalars, and Coordinate Systems.

2c. Explain the relationship between instantaneous and average values for physical quantities

• What is an instantaneous value? Give an example of an instantaneous value in physics.
• What is an average value? How is an average value calculated? Give an example of an average value in physics.

An instantaneous value is a value measured at a given instant, or time. For example, we can measure the velocity of an object at a given time as 5.5 km/s east. This is an instantaneous value because it was measured at a given instant. The velocity may not be constant over time, but at the instant it was measured, that was the velocity.

An average value is calculated over a period of time. For example, to calculate average speed, divide the distance traveled by time traveled. For example, if you drive 30 miles in two hours, your average speed is 15 miles/hour. However, as we know from driving, we rarely drive exactly the same speed for two hours. So, the instantaneous value of your speed could vary at any given time, but the average value is still 15 miles/hour.

2d. Compare and contrast speed and velocity

• Define elapsed time and show how it is calculated.
• Define average velocity and show how it is calculated.
• Why is velocity a vector quantity?
• Define instantaneous speed and average speed.
• Describe the velocity and speed for a given system.

Elapsed time, $\delta t$, is the change in time. Elapsed time is calculated in the following way:

$\Delta t=t_{f}-t_{i}$, where $t_f$ is final time and $t_i$ is initial time

The Greek letter delta, $\Delta$, means change. So, $\Delta t$ means change in time. You will see this frequently in this course. When calculating elapsed time, we often assume the initial time is zero, to make the subtraction easier.

Average velocity is the displacement divided by the elapsed time:

$\overline{v}\: =\: \frac{\Delta x}{\Delta t}\: =\: \frac{x_{f}-x_{i}}{t_{f}-t_{i}}$

Here, the line you see above the v shows that it is an average quantity. This is common notation for average quantities. To calculate the average velocity, divide the change in displacement by the elapsed time.

Review equations 2.5 and 2.6 in Average Velocity.

The average velocity is a vector quantity. This is because displacement is a vector quantity. Because we calculate average velocity from a vector quantity, it itself is a vector quantity. This means that average velocity has a direction associated with it. In one dimensional systems, this means that the average velocity is written with a (+) or (-) sign, depending on the direction of the displacement.

Instantaneous speed is the magnitude of the instantaneous velocity, measured at a given time or instant. Unlike velocity, instant speed is a scalar quantity, so it does not have a direction associated with it. For example, if the instantaneous velocity of an object is -2 m/s, the object's instantaneous speed is 2 m/s.

The average speed of an object, however, is not simply the magnitude of the average velocity. We define the average speed of an object as the distance divided by the elapsed time. Recall from learning outcome 2a that distance is a scalar quantity that describes how much an object moved and that it can be very different from the vector displacement. Therefore, the average speed of an object is also a scalar quantity, and it can differ from the average velocity.

Take a look at Figures 2.10 and 2.11. Figure 2.10 shows a diagram of the displacement and distance between a home and a store, which took 30 minutes total. For a roundtrip from home, to the store, and back home, the distance and displacement are quite different. The total distance traveled is 6 km (3 km to the store, and 3 km from the store back home). However, the total displacement is 0 because displacement is a vector. The person went +3 km to the store, and then -3 km back home. So, the total vector displacement is 0.

Based on the distance and displacement, we can calculate the average speed and velocity of the trip to the store and back. To determine the average speed, divide the distance by the elapsed time: 6 km/30 min = 2 km/min. However, to determine the average velocity, divide the displacement by the elapsed time: 0 km/30 min = 0 km/min. Figure 2.11 shows graphs of distance, average speed, and average velocity in this example.

2e. Solve one-dimensional kinematics problems

• Define acceleration, instantaneous acceleration, and average acceleration.
• Calculate displacement given average velocity and time.
• Calculate final velocity given initial velocity, acceleration, and time.
• Calculate the displacement of an accelerating object given acceleration and time.
• Calculate the final velocity of an accelerating object given acceleration and time.

Acceleration ($a$) is the rate of change of velocity. We can calculate the average acceleration using the following equation:

$\bar{a}=\frac{\Delta v}{\Delta t}=\frac{v_f-v_i}{t_f-t_i}$

Because velocity is a vector, acceleration is also a vector quantity.

Instantaneous acceleration is acceleration measured at a specific instant in time. In most kinematic problems, we assume average acceleration is a constant value.

We need to derive the equations of motion before we can calculate displacement given average velocity and time.

Review equations 2.28 and 2.29 in Solving for Displacement and Final Position from Average Velocity when Acceleration is Constant:

$x=x_0+\bar{v}t$

$\bar{v}=\frac{v_0+v}{2}$

where $x_0$ is initial displacement, $v_0$ is the initial velocity, and $v$ is the final velocity.

Review an example of using these equations to solve for displacement, given average velocity and time, in Example 2.8: Calculating Displacement: How Far Does the Jogger Run?.

We need to derive the equations of motion before we can calculate the final velocity given initial velocity, acceleration, and time.

Review equation 2.35 in Solving for Final Velocity:

$v=v_0+at$

where $v$ is final velocity, $v_0$ is initial velocity, $a$ is constant acceleration,
and $t$ is elapsed time.

Review an example of using these equations to solve for final velocity given initial velocity, acceleration, and time in Example 2.9: Calculating Final Velocity: An Airplane Slowing Down after Landing.

To calculate the final displacement of an accelerating object, we first need to derive the equation necessary to solve these problems.

Review equation 2.40 in Solving for Final Position when Velocity is Not Constant:

$x=x_0+v_0 t +\frac{1}{2}at^2$

where $x$ is final displacement, $x_0$ is initial displacement, $v_0$ is initial velocity, $t$ is elapsed time, and $a$ is acceleration.

Review an example using this equation to solve for final displacement given acceleration and time in Example 2.10: Calculating Displacement of an Accelerating Object: Dragsters.

To calculate the final velocity of an accelerating object, we need to derive the equation necessary to solve these problems.

Review the necessary equation 2.46 in Solving for Final Velocity when Velocity is Not Constant:

$v^2 = v_0^2 + 2a \left (x-x_0 \right )$

where $v$ is final velocity, $x$ is final displacement, $v_0$ is initial velocity, $a$ is acceleration, and $x_0$ is initial position.

In these problems, we first use equation 2.40 to solve for $x$, and then input that result into equation 2.46 to solve for final velocity.

Review an example using this equation to solve for final velocity in an accelerating object in Example 2.11: Calculating Final Velocity: Dragsters.

Review a list of the important kinematics equations used in this section in the box Summary of Kinematics Equations.

Review more examples of using kinematics equations in Example 2.12: Calculating Displacement: How Far Does a Car Go When Coming to a Halt? and Example 2.13: Calculating Time: A Car Merges into Traffic.

2f. Describe the effects of gravity on an object in motion

• Define gravity, free-fall, and acceleration due to gravity?
• How does air resistance affect the falling motion of objects on earth?

Gravity is a force that attracts objects toward the center of the earth, or another large object or planet. In the absence of friction or air resistance, all objects fall with the same acceleration toward the center of the earth. This is known as free-fall. The acceleration due to gravity, $g$, of an object in free-fall is $g=9.80\mathrm{\frac{m}{s^2}}$.

In reality, air resistance affects the acceleration of falling objects. Air resistance opposes the motion of an object in air, and causes falling lighter objects to accelerate less than heavier objects. This is why a feather falls to earth slower than a heavier object like a brick. If there was no air resistance, a feather and brick would fall to earth with the same acceleration due to gravity.

2g. Calculate the position and velocity of an object in free fall

• Calculate the position and velocity of a falling object given initial velocity and time.
• Use position and velocity data of an object in free fall to determine the acceleration due to gravity.

To perform calculations involving objects in free fall, we first need kinematic equations for objects in free fall. Fortunately, these equations are essentially the same as those found in learning outcome 2e above. The main difference is that in free fall, acceleration ($a$) equals the acceleration due to gravity ($g$). For an object falling, we use $-g$ to show the vector direction of free fall.

Review the relevant equations in the box Kinematics Equations for Objects in Free Fall where Acceleration = -g:

$v=v_0-gt$

$y=y_0+v_0 t-\frac{1}{2} gt^2$

$v^2 = v_0^2 -2g \left (y-y_0 \right )$

Note that because the motion is free fall, it is in the $y$ direction rather than the $x$ direction. Here, $g$ is acceleration due to gravity, $g=9.80\mathrm{\frac{m}{s^2}}$.

When calculating the position and velocity of a falling object, we need to consider two different conditions. First, the object can be thrown up and then enter free fall. For example, you could throw a baseball up and watch it fall back down.

Review this case in Example 2.14: Calculating Position and Velocity of a Falling Object: A Rock Thrown Upward. After reviewing the solution, pay special attention to the graphs in Figure 2.40 in the example.

The other case is when an object is thrown directly downward. For example, you could throw a baseball directly down from a second-floor window.

Review this case in Example 2.15: Calculating Velocity of a Falling Object: A Rock Thrown Down. After completing this example, review Figure 2.42, which compares what is happening in Examples 2.14 and 2.15. It is important to understand the difference between an object that is thrown up and enters free fall, versus an object that is directly thrown down.

We can often use experimental data to calculate constants, such as gravity. In Example 2.16: Find G from Data on a Falling Object the acceleration due to gravity constant ($g$) is determined from experimental data.

2h. Draw and interpret graphs for displacement and velocity as functions of time, and determine velocity and acceleration from them

• Define the dependent and independent variable in a graph.
• Describe the graph of a straight line.
• Analyze a graph of position versus time when acceleration is zero.
• Analyze a graph of position versus time when acceleration is constant (not zero).

In graphing two variables against each other, we define the dependent variable as the variable on the vertical axis (y–axis) and the independent variable as the variable on the horizontal axis (x–axis). When plotting a line, we use the equation $y = mx + b$, where $m$ is the slope and $b$ is the y–intercept of the line.

We define slope as:

$m=\mathrm{\frac{rise}{run}}=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

The y–intercept is the point where the line crosses the y–axis of the graph.

Review Figure 2.46.

An example of a linear graph is the graph of position versus time when acceleration is zero.

Review an example of this type of graph in Figure 2.47. In this graph, we can determine the slope by picking two different points and calculating slope using the equation above. In this case, the unit for slope is m/s, which is the unit for velocity. Therefore, the slope for a graph of position versus time with zero acceleration is the average velocity of that object.

Review how to calculate the average velocity of an object from this type of graph in Example 2.17: Determining Average Velocity from a Graph of Position versus Time: Jet Car.

When acceleration is a non-zero constant, the graph of position versus time is no longer linear. Review an example of this type of graph in Figure 2.48.

Note that while the position versus time graph is not linear, the velocity versus time graph is linear. In the position versus time graph, the slope at any given point is the instantaneous velocity of the object. The instantaneous slope can be determined by drawing tangent lines at various points along the graph, and using the tangent lines to determine slope.

Review tangent lines drawn in Figure 2.48 (a).

To determine instantaneous velocity at a given time when acceleration is a non-zero constant, review Example 2.18: Determining Instantaneous Velocity from the Slope at a Point: Jet Car. 

We can determine instantaneous velocity at multiple points along a position-time graph with constant non-zero acceleration. Then, we can plot velocity versus time, seen in Figure 2.48 (b). We see that this is a linear graph. The slope has units of m/s², which are acceleration units. Therefore, the slope of the velocity versus time graph is acceleration.

Unit 2 Vocabulary

• Acceleration
• Acceleration due to gravity
• Air resistance
• Average acceleration
• Average speed
• Average value
• Average velocity
• Delta
• Dependent variable
• Displacement
• Displacement of an accelerating object
• Distance
• Equations of motion
• Elapsed time
• Final velocity of an accelerating object
• Free fall
• Gravity
• Independent variable
• Instantaneous acceleration
• Instantaneous speed
• Instantaneous value
• Position
• Scalar physical quantity
• Scalar quantity
• Slope
• Speed
• Tangent line
• Vector physical quantity
• Vector quantity
• Velocity
• Velocity of a falling object
• Y–axis
• Y–intercept