Motion in Two Dimensions Study Guide (page 2)
In this lesson, we will continue the study of important motion quantities and learn about motion in two dimensions, relative velocity, and projectile motion.
A vector can represent behavior over three separate directions x, y, and z. The same concepts of displacement, average and instantaneous velocity, and average and instantaneous acceleration will apply to motion in two dimensions. The major difference is that we will work with formulas containing only one variable at a time: x or y or z. This is a real advantage, as in separate directions, different interactions might predominate, and therefore, the source of interaction (and the specific force) will be considered for separate directions.
In this case, the vector position for each of the initial and final positions is determined by two coordinates: x and y.
Δ x = xf – xi
Δ y = yf – yi
And from these components, the displacement of the vector is:
Δ r = (Δ x, Δ y)
Δ r = Δ x + Δ y
The displacement vector is the segment that connects the initial and final positions, and is directed toward the final position:
Δ r = rf – bi
A toy car starts from the corner of the classroom and travels to a new position of coordinates (5,6) meters away from the corner. Draw the displacement and find the magnitude.
We can see that the two measurements in the problems are expressed in meters, so no conversion is necessary. We will draw a system of coordinates, and we will place the car in the origin of the system at the beginning of motion. This is the initial position. The final position is given by the problem. We will construct the displacement as the vector that connects the initial and final position and then find the magnitude by using the components.
Now the problem becomes a one-dimensional problem because we can write the equations for motion on the separate axis (see Figure 4.1):
(xi, yi) = (0,0) m
(xf, yf ) = (5,6) m
Δ x = xf – xi
Δ x = 5 – 0 = 5 m
Δ y = yf – yi
Δ y = 6 – 0 = 6 m
Δ r = (5,6) m
Δ r = (52 + 62)1/2 = 8 m
If the motion is more complicated (for example, a nonlinear trajectory), then we would have cases where the distance traveled is not necessarily the same as the displacement. For example, in the case shown in Figure 4.2, the distance is larger than the displacement.
A car travels 40 miles to the west and then another 30 miles to the north. Draw the displacement, calculate the magnitude and direction of the displacement, and compare to the distance traveled.
As the single values of this problem are in miles and no other requirement exists for conversion in the problem, we can keep the solution in miles. If we are asked to convert to SI, we would multiply each of the two distances with the conversion factor 1,609 meters/1 mile to determine the result in meters.
Δ ri = 40 miles W
Δ rf = 30 miles N
Δ rtotal = ?
distance = ?
The distance traveled by the car is the sum of the two distances:
Distance = 40 miles + 30 miles = 70 miles
The total displacement has to take into account each of the displacements, but we cannot simply add the two numbers because their vector expressions put them in different directions. Let us draw a motion diagram. We will consider east as the positive x-axis and north as the positive y-axis (see Figure 4.3).
(Δ rtotal)2 = (Δ ri )2 + (Δ rf )2
Δ rtotal = [(Δ ri)2 + (Δ rf)2]1/2
Δ rtotal = 50 miles
The direction is the angle θ, and it can be calculated from the geometry of the problem:
sin θ = rf /rtotal = 30 miles/50 miles
θ = sin–1 (0.6) = 37°
Hence, displacement is Δ rtotal = 50 miles at 37° north of west or (40 miles,30 miles) by the x and y components.
It can be seen that the distance is larger compared to the displacement, as shown in the figure (the two sides of the triangle added together are larger than the hypotenuse).
Based on the definition of displacement, we introduce also the average velocity.
When the time interval is very small (approaching zero, or Δ t→ 0), the definition of the average velocity becomes the definition of the instantaneous velocity.
If the motion is such that velocity is constant (both in magnitude and direction), then the average speed and the instantaneous velocities are the same.
Average velocity is the time rate of change of displacement:
A toy car starts moving at a constant speed for 10 seconds. Show with vectors, at two separate positions, the average and instantaneous velocities.
We will represent the car in a two-dimensional motion by using a point-like object and drawing the two velocities. They are equal vectors (this means that both direction and magnitude are the same). See Figure 4.4.
When the trajectory is nonlinear, instantaneous velocity is a vector always tangent to the trajectory as shown in Figure 4.5.
Instantaneous velocity is velocity at a certain time:
Use a diagram to show the displacement of a car that moves between the following positions: (50,30) meters to (–40,40) meters in 4 seconds, and then calculate displacement and the average velocity, both expressed with their coordinates and magnitude and angle.
All units are expressed in SI. In order to calculate the average velocity, we need to determine displacement. Figure 4.6 shows the initial and final positions.
Δx = xf – xi
Δx = – 40 – 50 = – 90 m
Δy = yf – yi
Δy = 40 – 30 = 10 m
Δr = (–90,10) m
Δr = (902 + 102)1/2 = 91 m
tan θ = Δy/Δx = 10 m/( –90 m) = –1/9
θ = tan–1(1/9) = –6°
Therefore, displacement is 91 meters at an angle of 6° north of west.
Average velocity is:
Vaverage = (22.52 + 2.52)1/2 = 23 m/s
θ = tan–1 (–2.5/22.5) = –6°
Or average velocity is 23 m/s (also = 91 m/4 s) at an angle of 6° north of west.
We can see that the average velocity and displacement are in the same direction. As noted previously, the instantaneous velocity will be tangent to the trajectory at every point but not necessarily in the same direction as displacement and average velocity.
If the motion is made with changing velocity, then acceleration can be defined.
Similar to the average and instantaneous velocity, starting with the definition just given, when the time interval approaches zero, one can define instantaneous acceleration.
As seen in both these definitions, the vector acceleration has the direction of the change in velocity, and that can be determined by either change in magnitude or change in direction.
Average acceleration is the time rate of change of velocity:
Consider an object moving on a curve as shown in Figure 4.7. Draw the vector average acceleration based on the information in the graph.
Start by constructing the vector Δv and then draw the average acceleration aaverage based on this direction.
Because average acceleration is nothing else, then by dividing a vector (Δv) by a scalar (Δt), there will be no change in direction. Only magnitude will be increased or decreased, depending on the size of the denominator.
Instantaneous acceleration is acceleration at a certain time:
As you have probably started to notice, a lot in mechanics is relative. All measurements start with distance and time, neither of which have an absolute origin that can be easily applied to all sorts of motion. So, what solution do we have for this issue? The answer is to consider our own reference frames and origins of space and time. You have considered a simple, one-axis system of reference in linear motion. In this chapter, we worked on a reference system composed of two perpendicular axes: x and y. In real space, a need exists for a third axis—height, or the z-axis. And even more, some consider that time be represented by a fourth coordinate, because motion spans not only in space but also in time. Depending on the axes of reference, motion looks different. Imagine yourself in space, far away from any cosmic object. Is your ship moving or not? If the engines are quiet, and there is no acceleration, you will not be able to tell your motion if you are not looking at your senso. But if an alien species is watching you with deep space sensors from a far away planet, they will be able to tell if you are moving and how are you moving. So, who is right? Sure enough, the answer is both observants: the alien species and you. The difference is that each of you relates to different frames: Your frame moves with you, so evidently, you cannot determine whether you are at rest or not, while the aliens analyze your motion relative to a different system and see that you are in motion with respect to their system. This is why for more down-to-earth motion, we define relative velocity.
The fixed frame reference for most of our problems will be Earth. We will not consider Earth's motion around its own axis or around the sun when we are considering an apple falling from a tree or a car or boat traveling for some distance. In other cases, such as for space travel, a distant star can be considered as a fixed position. Although this is not completely true (the distant star is moving), if the stellar object is far enough away, the motion during the experiment might be negligible compared to the motion of the analyzed system.
Relative velocity is the resultant vector measuring the velocity of an object in a fixed frame.
vr = v1 – v2
Where v1 is the speed of an object with respect to a moving frame and v2 is the speed of the frame with respect to a fixed frame.
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