Concept Summary

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Topic | Discussion | Learning Aids
Displacement vector | 3.1 DISPLACEMENT, VELOCITY, AND ACCELERATION The position of an object is located with a vecto drawn from the coordinate origin to the object. The displacement of the object is defined as , where and specify its final and initial positions, respectively. |
Average velocity | The average velocity of an object moving between two positions is defined as its displacement divided by the elapsed time :

| (3.1)

Instantaneous velocity | The instantaneous velocity is the velocity at an instant of time. The average velocity becomes equal to the instantaneous velocity in the limit that becomes infinitesimally small ():

Average acceleration | The average acceleration of an object is the change in its velocity, divided by the elapsed time :

| (3.2)

Instantaneous acceleration | The instantaneous acceleration is the acceleration at an instant of time. The average acceleration becomes equal to the instantaneous acceleration in the limit that the elapsed time becomes infinitesimally small:

| 3.2 EQUATIONS OF KINEMATICS IN TWO DIMENSIONS Motion in two dimensions can be described in terms of the time t and the x and y components of four vectors: the displacement, the acceleration, and the initial and final velocities. |
Independence of the x and y parts of the motion | The x part of the motion occurs exactly as it would if the y part did not occur at all. Similarly, the y part of the motion occurs exactly as it would if the x part of the motion did not exist. The motion can be analyzed by treating the x and y components of the four vectors separately and realizing that the time t is the same for each component. |
Equations of kinematics for constant acceleration | When the acceleration is constant, the x components of the displacement, the acceleration, and the initial and final velocities are related by the equations of kinematics, and so are the y components:

x Component |  | y Component |
| 3.3a |  | 3.3b
| 3.4a |  | 3.4b
| 3.5a |  | 3.5b
| 3.6a |  | 3.6b

The directions of these components are conveyed by assigning a plus (+) or minus (−) sign to each one.
Acceleration in projectile motion | 3.3 PROJECTILE MOTION Projectile motion is an idealized kind of motion that occurs when a moving object (the projectile) experiences only the acceleration due to gravity, which acts vertically downward. If the trajectory of the projectile is near the earth’s surface, ay has a magnitude of 9.80 m/s2. The acceleration has no horizontal component (), the effects of air resistance being negligible. |
Symmetries in projectile motion | There are several symmetries in projectile motion: (1) The time to reach maximum height from any point is equal to the time spent returning from the maximum height to that point. (2) The speed of a projectile depends only on its height above its launch point, and not on whether it is moving upward or downward. |

Use Self-Assessment Test 3.1 to evaluate your understanding of Sections 3.1–3.3.

Adding relative velocities | 3.4 RELATIVE VELOCITY The velocity of object A relative to object B is , and the velocity of object B relative to object C is . The velocity of A relative to C is (note the ordering of the subscripts)

While the velocity of object A relative to object B is , the velocity of B relative to A is .

Use Self-Assessment Test 3.2 to evaluate your understanding of Sections 3.4.

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