### SAT Physics Kinematics in Two Dimensions - Independence Of Motion

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**SAT Physics Kinematics in Two Dimensions - Independence Of Motion**

**INDEPENDENCE OF MOTION**

Chapter 3 demonstrated how kinematic equations are used determine the position, velocity, and acceleration of an object moving along a one-dimensional line. Consider an example of

a straight line. The diagram below indicates the instantaneous velocity vectors on the ball at four different locations during its motion.

**Figure 4.1.**Velocity vectors

Consider if someone throws the same ball horizontally near the surface of Earth. If the ball

**Figure 4.2.**Projectile velocity vectors

Kinematic equations can be used for motion only along a straight line. Therefore, separate kinematic equations must be employed for x-variables and for y-variables. The resulting motion is described by two kinematic equations in combination. Adding x- and y-subscripts to the kinematic variables allows you to distinguish between similar variables acting in different directions. Table 4.2 compares the kinematic equations in one and in two dimensions.

**Table 4.2**Kinematic Equations in One and Two Dimensions

**TRUE VELOCITY AND DISPLACEMENT**

The kinematic equations solve for x- and y-direction velocities and displacements. However, in two-dimensional motion problems, the path followed by the object does not lie solely along either the x- or y-axis. The ball in Figure 4.3 follows a parabolic path.

**Figure 4.3.**Projectile motion velocity vectors and their components

**RELATIVE VELOCITY**

The motion of an object, as described by two observers, may differ depending on the location of the observers. For example, a car reported as moving at 30 meters per second by a stationary observer will appear to be moving at 10 meters per second as observed by a driver in a car traveling alongside at 20 meters per second. Problems involving multiple velocities are known as relative velocity problems. Common two-dimensional relative velocity problems involve a boat moving across a river or an airplane flying through the air. In these problems, the velocities in both the x- and y-directions are constant. Therefore, the acceleration in the

**Table 4.3**Relative Velocity Equations

**Determinig True Velocity**

**(A)**Determine the magnitude of the true velocity of the plane with respect to an observer on the ground.

**WHAT'S THE TRICK?**

An observer on the ground will see the true velocity of the airplane. Mathematically, this is the resultant vector created by adding the airplane and wind velocity vectors tip to tail, as shown below, and using the Pythagorean theorem.

**(B)**Which vector, of the choices given below, describes the direction the pilot must aim the plane in order for the plane to have a true velocity that points directly north?

**WHAT'S THE TRICK?**

The velocity vectors for the plane and the wind must add together to create a true velocity that points north. Since the wind is blowing out of the west, the plane must have a component that moves toward the west to cancel the effect of the wind.

**(i)**is the correct heading for the plane. When the plane is aimed northwest, it will actually move with a true velocity directly north.

**Determining True Displacement**

**WHAT'S THE TRICK?**

The boat and the river both move at constant velocity. The motion of the boat and the river are mathematically independent but take place simultaneously. You should include subscripts to distinguish between the two velocities. The velocity of the boat, v

_{b}, produces a cross-stream displacement, x. The velocity of the river, v

_{r}produces a downstream displacement, y.

**PROJECTILE MOTION**

Projectile motion describes an object that is thrown, or shot, in the presence of a gravity field. An object is considered a projectile only when it is no longer in contact with the person, or device, that has thrown it and before it has come into contact with any surfaces. As a result, the downward acceleration of gravity is the only acceleration acting on a projectile during its flight.

One of the most important aspects of projectile motion is the type of motion experienced in each direction. If the vertical acceleration of gravity is the only acceleration acting on a projectile, the horizontal speed of a projectile cannot change. Therefore, the horizontal component of velocity must always remain constant. Both the vertical velocity and the vertical displacement will be affected by the acceleration of gravity. When calculating the vertical portion of projectile motion, you must use the complete kinematic equations, as shown in Table 4.4. You should include additional subscripts to distinguish the and velocities from

**Table 4.4**Kinematic Equations with Gravity

**Table 4.5**Velocity of Two Typical Launches

**Horizontally Launched Projectiles**

Horizontally launched projectiles are the most common projectile motion problems encountered on introductory physics exams. As with any kinematics problem, identifying variables (especially hidden variables) is extremely important. In horizontal launches, the initial velocity in the y-direction is zero as shown in Table 4.6. This simplifies the y-direction equations.

**Table 4.6**Kinematic Equations for Horizontal Launches

**Horizontally Launched Projectiles**

**WHAT'S THE TRICK?**

First determine the x- and y-components of the initial velocity. This is easy for horizontal launches. All of the initial velocity is directed horizontally, and none is directed vertically.

If time is unknown, solve for time using y-direction equations. The vertical displacement of 5 meters is given. Use an equation containing both displacement and time.

**Projectiles Launched at an Angle**

Projectiles launched at angles are more difficult to solve mathematically. As a result, complex calculations for these projectiles may not appear on the SAT Subject Test in Physics. However, projectiles launched at upward angles do have several unique characteristics that will be tested conceptually.

**Figure 4.4.**Projectile motion vectors

Examination of the diagram reveals four key facts about projectiles launched at angles.

- The horizontal component of velocity, v
_{x}, remains constant. - When the projectile is moving upward, its vertical speed decreases by 10 meters per second every second until the projectile reaches an instantaneous vertical speed of zero at maximum height. The decreasing velocity then results in a changing downward speed that increases by 10 meters per second every second.
- The projectile passes through each height twice, once on the way up and once on the way down (except the single point at maximum height). At points with equal height, the magnitude of the vertical velocity on the way up equals the magnitude of the vertical velocity on the way down.
- The time the projectile rises equals the time the projectile falls, as long as the final height equals the initial height.

**Figure 4.5.**Two key instants during a projectile flight

- The vertical component of velocity at maximum height is zero. Therefore, the true velocity at maximum height equals the horizontal component of the launch velocity:

- Although the vertical component of velocity at maximum height becomes zero, the acceleration in the vertical direction remains a constant -10 meters per second squared
- For a projectile landing at its initial launch height, the time to maximum height is half the total time of flight.

- Any two launch angles totaling 90 degrees will have the same range. For example, if a projectile is launched at 30 degrees, it will reach the same impact point if it is launched at 60 degrees.

- The vertical component of velocity at maximum height is zero. Therefore, the true velocity at maximum height equals the horizontal component of the launch velocity:

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