Topic 03: Linear Kinematics (Part 1)

Topic 03: Linear Kinematics (Part 1)

Lesson Learning Outcomes

• Distinguish between linear, angular, and general motion
• Define distance traveled and displacement and distinguish between the two
• Define average speed and average velocity and distinguish between the two
• Define instantaneous speed and instantaneous velocity
• Name the units of measurement for distance traveled and displacement, speed and velocity.

Guiding Questions

• Why do we need to classify motion?
• Why is it important to distinguish distance and displacement?
• Why is it important to distinguish between average speed and average velocity
• Why is speed and velocity important in sport activities?

DEFINITION OF MOTION

Motion is defined as the action or process of a change in position. Movement is a change in position. Moving involves a change in position from one point to another. Two things are necessary for motion to occur: space and time—space to move in and time during which to move. To make the study of movement easier, we classify movements as linear, angular, or both (general).

1. Linear Motion

Linear motion is also referred to as translation. It occurs when all points on a body or object move the same distance, in the same direction, and at the same time.This can happen in two ways: rectilinear translation or curvilinear translation.

Rectilinear translation occurs when all points on a body or object move in a straight line so that the direction of motion does not change, the orientation of the object does not change, and all points on the object move the same distance.

Curvilinear translation occurs when all points on a body or object move so that the orientation of the object does not change and all points on the object move the same distance.

2. Angular Motion

Angular motion is also referred to as rotary motion or rotation. It occurs when all points on a body or object move in circles (or parts of circles) about the same fixed central line or axis. Angular motion can occur about an axis within the body or outside of the body.

To determine whether or not a motion is angular, imagine any two points on the object in question. As the object moves, are the paths that each of these points follow circular? Do these two circular paths have the same center or axis? If you imagine a line connecting the two imaginary points, does this line continuously change orientation as the object moves? Does the line continuously change the direction in which it points? If these conditions are true, the object is rotating.

3. General Motion

General motion is a combination of linear and angular motions. General motion is the most common type of motion exhibited in sports and human movement. Running and walking are good examples of general motion. In these activities, the trunk often moves linearly as a result of the angular motions of the legs and arms. Bicycling is another example of general motion.

Linear Kinematics

Position

Our definition of motion—the action or process of change in position—refers to position. Mechanically, position is defined as location in space.

The Cartesian coordinate system helps to identify position.

First, we would need to locate a fixed reference point for our coordinate system. This fixed point is called the origin, because all our position measurements originate from it. With this system, we could identify a position in two dimensions with two numbers corresponding to x- and y-coordinates

Distance Travelled

Distance travelled is easily defined—it’s simply a measure of the length of the path followed by the object whose motion is being described, from its starting (initial) position to its ending (final) position.

Distance travelled doesn’t mean a whole lot though, because the direction of travel isn’t considered.

Displacement

Displacement is the straight-line distance in a specific direction from initial (starting) position to final (ending) position. The resultant displacement is the distance measured in a straight line from the initial position to the final position. Displacement is a vector quantity. As you recall, force was also a vector quantity. A vector has a size associated with it as well as a direction. It can be represented graphically as an arrow whose length represents the size of the vector and whose orientation and arrowhead represent the direction of the vector. Representation of displacement with an arrow is appropriate and communicates what displacement means as well.

Distance Travelled vs Displacement: Example

Suppose a football player has received the kickoff at his 5 yd line, 15 yd from the left sideline. His position on the field (using the Cartesian coordinate system we established in the previous section) is (15, 5) when he catches the ball. He runs the ball back following the path shown in figure a. He is finally tackled on his 35 yd line, 5 yd from the left sideline. His position on the field at the end of the play is (5, 35). If we measure the length of the path of his run with the ball, it turns out to be 48 yd. So we might describe this run as a run of 48 yd to gain 30 yd.

Figure b shows the path of the player in the kick return example. The arrow from the initial position of the player to where he was tackled represents the displacement of the back.

If we resolve this resultant displacement into components in the x-direction (across the field) and y-direction (down the field toward the goal), we then have a measure of how effective the run was. In this case, the y-displacement of the running back is the measure of importance. His initial y-position was 5 yd and his final y-position was 35 yd. We can find his y-displacement by subtracting his initial position from his final position:

d_y = 𝚫y = y_f − y_i

where

d_y = displacement in the y-direction,

𝚫 = change, so 𝚫y = change in y-position,

y_f = final y-position, and

y_i = initial y-position.

If we put in the initial (5 yd) and final (35 yd) values for y-position, we get the runner’s y-displacement:

d_y = 𝚫y = y_f − y_i = 35 yd − 5 yd

d_y = +30 yd

We can do the same for the displacement across the field (in the x-direction)

If we put in the initial (15 yd) and final (5 yd) values for x-position, we get the runner’s x-displacement:

d_x = 𝚫x = x_f − x_i = 5 yd − 15 yd

d_x = −10 yd

We could find the resultant displacement of the runner similarly to the way we found a resultant force. Graphically, we could do this by drawing the arrows representing the component displacements of the runner in the x- and y-directions. Put the tail of the x-displacement vector at the tip of the y-displacement vector, and then draw an arrow from the tail of the y-displacement vector to the tip of the x-displacement vector. This arrow represents the resultant displacement. We could also determine this resultant displacement using trigonometric relationships

(𝚫x)² + (𝚫y)² = R².

For our displacements, then, we can substitute −10 yd for 𝚫x and +30 yd for 𝚫y and then solve for R, which represents the resultant displacement.

(−10 yd)² + (30 yd)² = R²

1000 yd² = R²

R = √1000 yd² = 31.6 yd

To find the direction of this resultant displacement, we can use the relationship between the two sides of the displacement triangle.

Distance travelled can be described by a single number that represents the length of the path followed by the object during its motion. Displacement, however, is a vector quantity, so it is expressed with a length measurement and a direction. The resultant displacement is the length of a straight line from the initial position to the final position in the direction of motion from the initial position to the final position. Components of the resultant displacement may also be used to describe displacement of the object in specific directions

Speed

Speed is the rate of motion. More specifically, it is the rate of distance traveled. It is described by a single number only.

Average speed of an object is distance traveled divided by the time it took to travel that distance. Mathematically, this can be expressed as

The units for describing speed are a unit of length divided by a unit of time. The SI unit for describing speed is meters per second (m/s).

The speed of an object at a specific instant of time is its instantaneous speed. The speed of an object may vary with time, especially in an event such as a 100 m dash. The maximum or top speed a runner achieves during a race is an example of an instantaneous speed. An average speed gives us an estimate of how fast something was moving over only an interval of time—not an instant in time.

We can think of instantaneous speed as distance traveled divided by the time it took to travel that distance if the time interval used in the measurement is very small.

Velocity

Average velocity is displacement of an object divided by the time it took for that displacement. Because displacement is a vector, described by a number (magnitude) and a direction, average velocity is also a vector, described by a number (magnitude) and a direction. Mathematically, this can be expressed as

The SI unit for describing velocity is meters per second. To measure the average velocity of an object, you need to know its displacement and the time taken for that displacement.

Sometimes we are interested in the components of velocity. So, just as we were able to resolve force and displacement vectors into components, we can also resolve velocity vectors into components. To resolve a resultant average velocity into components, we could simply determine the components of the resultant displacement.

For the football player returning the kickoff in the example used earlier, the player’s displacement from the instant he received the ball until he was tackled was −10 yd in the x-direction (across the field) and +30 yd in the y-direction (down the field). His resultant displacement was 31.6 yd down and across the field (or −71.6° across the field). If this kick return lasted 6 s, his resultant average velocity was

This resultant average velocity was in the same direction as the resultant displacement. Similarly, the running back’s average velocity across the field (in the x-direction) would be the x-component of his displacement divided by time or

The running back’s average velocity down the field (in the y-direction), which is the most important of all these velocities, would be the y-component of his displacement divided by time or

Just as with the displacements, the resultant average velocity is larger than any of its components. And just as with the displacements, the square of the resultant average velocity should equal the sum of the squares of its components.

This indeed matches the resultant average velocity of 5.3 yd/s we computed from the resultant displacement and elapsed time.

Generally, if the motion of the object under analysis is in a straight line and rectilinear, with no change in direction, average speed and average velocity will be identical in magnitude (as in the case of a 100m dash event). However, if we are speaking of an activity in which the direction of motion changes, speed and the magnitude of velocity are not synonymous.

Instantaneous velocity is the velocity of an object at an instant in time. When we speak of the magnitude of the resultant instantaneous velocity of an object, that number is the same as the instantaneous speed of the object. A resultant instantaneous velocity can also be resolved into components in the direction of interest. For the football player running back the kick-off, we could describe his instantaneous resultant velocity, and we could also describe his instantaneous velocity in the x-direction (across the field) or the y-direction (down the field).

Importance of Speed and Velocity

In 2010, Ardolis Chapman of the Cincinnati Reds threw a fastball pitch that was clocked at 154 ft/s or 47 m/s. The distance from the pitching rubber to home plate is 60 ft 6 in., or 60.5 ft (18.4 m). The ball is released about 2 ft 6 in. in front of the rubber, so the horizontal distance it must travel to reach the plate is only 58 ft (60.5 ft − 2.5 ft) or 17.7 m. Another way to say this is that the horizontal displacement of the ball is 58 ft. How much time does a batter have to react to a fastball pitched at 105.1 mi/h? If we assume that this is the average horizontal velocity of the ball during its flight, then, using equation

A batter only has 0.38 s to decide whether or not to swing his bat, and if he does decide to swing it, he has to do so in the time he has left. The faster the pitcher can pitch the ball, the less time the batter has to react, and the less likely it is that the batter will hit the ball

Summary

• Motion may be classified as linear, angular, or a combination of the two (general motion).
• Linear distance represents the length of the path followed form start to finish.
• Linear displacement is the straight-line distance from start to finish.
• Speed is the rate of change of distance whereas velocity is the rate of change of displacement.The fundamental dimensions used in mechanics are length, time, and mass. The SI units of measurement for these dimensions are the meter (m) for length, the second (s) for time, and the kilogram (kg) for mass.
• Displacement and velocity are vector quantities which need to be described by both size and direction.