Motion in Two Dimensions Study Guide (page 2)

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Solution 3

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.

Two-Dimensional Motion

When the trajectory is nonlinear, instantaneous velocity is a vector always tangent to the trajectory as shown in Figure 4.5.

Two-Dimensional Motion

Instantaneous Velocity

Instantaneous velocity is velocity at a certain time:

Example 4

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.

Solution 4

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.

Two-Dimensional Motion

Δx = xfxi

Δx = – 40 – 50 = – 90 m

Δy = yfyi

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

Average acceleration is the time rate of change of velocity:

Example 5

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.

Two-Dimensional Motion

Solution 5

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

Instantaneous acceleration is acceleration at a certain time:

Relative Motion

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

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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