# Kinematics

Kinematics is the study of motion. In kinematics, you will learn about both one-dimensional and two-dimensional motion as they relate to displacement, velocity, and acceleration. You will also be acquainted with the Big 5, a set of five equations that are extremely important in physics.

Content that appears in a box similar to this is content that applies only to the AP C curriculum. AP B and AICE students can skip any content contained in these boxes.

### Motion in One Dimension

- Displacement and Velocity
- It is important to define a frame of reference so that it is known what an object is moving with respect to. For example, a passenger sitting in a moving car is not moving with respect to the car, however they are moving with respect to stationary landmarks outside the car.
- Displacement
- Displacement is the straight-line distance from the initial position to the final position.
- Displacement is NOT always the same as distance traveled. For example, if a ball is rolled 18 meters, but hits a wall and rolls back 3 meters, the distance traveled is 21 meters, but the displacement is only 15 meters.
- Positions are represented with an
*italicized*lowercase letter "*x*." An initial position may be be denoted as*x*or_{i}*x*. A final position is usually denoted as_{0}*x*._{f} - Displacement is often represented as "
*Δx*," pronounced "delta x." Delta is a greek letter used by modern scientists to represent a change, so*Δx*is literally "the change in x." - In physics, displacement is measured in meters, m.

- Average Velocity
- Velocity is a vector quantity, meaning it has both a magnitude and a direction. Speed is a scalar quantity, meaning it has only a magnitude. So, velocity is the speed in a particular direction.
- Velocity and speed are usually represented by an
*italicized*lowercase letter "*v*." - Average velocity,
*v*_{avg}, is calculated by the following equations: (*x*-_{f}*x*)/(_{i}*t*-_{f}*t*), or_{i}*Δx*/*Δt*. Note that*t*represents time. - In physics, time is measured in seconds, s, and velocity is measured in meters per second, m/s.

- Average Speed
- Though one might think that average speed would just be the magnitude of the average velocity, this is not the case.
- In reality, average speed is the total distance traveled,
**not**the displacement, divided by the time interval. - You will see how this can make a major difference in the practice problems for kinematics.

- Instantaneous Velocity
- Where average velocity is over an interval, instantaneous velocity is the velocity at one given time or position.
- Instantaneous velocity can be found by using a velocity function, such as this equation given constant acceleration
*a*, time*t*, and initial velocity,*v*:_{i}*v*=*at*+*v*._{i} - A velocity function
*v*(*t*) can be found by taking the derivative of any position function*x*(*t*). You can also integrate to go from the velocity function to the position function.

- Acceleration
- Acceleration,
*a*, simply measures any change in an object's velocity. Since velocity is a vector quantity, acceleration is also a vector, so it has both a magnitude and a direction. - If an object's speed increases while moving in the positive direction, its acceleration is positive. If the speed decreases while moving in the positive direction, the acceleration is negative. If the speed increases while moving in the negative direction, the acceleration is negative. If the speed decreases while moving in the negative direction, the acceleration is positive.
- Since acceleration is any change in an object's velocity, which is a vector, that means that a moving object whose direction is changing is experiencing an acceleration.
- If an object is moving at a constant velocity, it is said to have no acceleration,
*a*= 0. - In physics, acceleration is measured in meters per second per second, m/s
^{2}. - Average Acceleration
- Average acceleration,
*a*, is similar to average velocity in that it is measured over an interval._{avg} *a*= (_{avg}*v*-_{f}*v*)/(_{i}*t*-_{f}*t*) =_{i}*Δv*/*Δt*

- Average acceleration,
- Instantaneous Acceleration
- Similarly to instantaneous velocity, instantaneous acceleration is the acceleration at a given time or position, and can be calculated using an equation containing acceleration.
- An acceleration function
*a*(*t*) can be found by taking the derivative of any velocity function*v*(*t*). You can also integrate to go from the acceleration function to the velocity function.

- Constant Acceleration
- Constant acceleration occurs when an object's velocity changes by the same amount per unit time. In this case, the object's acceleration can be defined simply as a number.
- The most common and important example of constant acceleration is an object in free fall, discussed below.
- The following five equations all deal with the five kinematics quantities we've covered (
*x*,*v*,_{i}*v*or*v*,_{f}*a*, and*t*) when the acceleration is constant. These equations are known as the Big 5, and are very important for any AP or AICE physics student.*Δx*=^{1}/_{2}(*v*+_{i}*v*)_{f}*Δt**v*=*at*+*v*_{i}*x*=^{1}/_{2}*at*^{2}+*v*+_{i}t*x*_{i}*x*=*x*+_{i}*vt*-^{1}/_{2}*at*^{2}*v*^{2}=*v*_{i}^{2}+ 2*a*(*x*-*x*)_{i}- Note that the first and fourth equations are not as important as the other three, but nonetheless it is highly recommended that any AP or AICE physics student memorize all of the Big 5. Also notice that the third and fourth look very similar if the terms were put in the same order, the differences being the sign of
^{1}/_{2}*at*^{2}and the fact that the third uses*v*while the fourth uses_{i}*v*(for these equations,*v*is the same as*v*)._{f} - Note that the second equation is the derivative of the third with respect to t, assuming all other values are constant. Because of this, it is possible for a calculus student to memorize only four of the Big 5, and derive the second from the third when necessary. Also note that the derivative of the second equation with respect to t is
*a*(*t*) =*a*, a constant.

- Acceleration,
- Objects in Free Fall
- If an object is in free fall, this means that the object is being influenced only by gravity. Realistically, this is not possible because of air resistance, but for AP and AICE physics, we assume that air resistance is negligible, making free fall possible.
- The effects of gravity cause an object in free fall to experience a constant acceleration, called gravitational acceleration, which is represented by the letter
*g*. On Earth, the value of*g*is known to be 9.81 m/s^{2}downward, which is generally considered the negative direction. Note that this can be rounded to 10 m/s^{2}when necessary to simplify calculations. - Since an object in free fall experiences a constant acceleration, the Big 5 equations can be applied to describe the object's motion. Note that any
*x*in the big five can be replaced with a*y*for vertical motion, but in this case all*x*'s must be changed to*y*'s. - Also note that the acceleration due to gravity is the same regardless of the object's mass. With air resistance, this would not be entirely true because of differences in surface area, but this can be ignored on the AP and AICE exams.

### Two-Dimensional Motion

- Projectile Motion
- In physics, a projectile is defined as any object that moves through the air in a parabolic arc under only gravitational acceleration. An example of a projectile, assuming negligible air resistance, would be any object thrown through the air, such as a basketball.
- Projectiles move in two directions, vertical and horizontal. These two directions of motion can be treated seperately for calculations.
- Vertical Motion
- The vertical motion of a projectile is mathematically the same as the motion of an object in free fall, so the Big 5 can be applied in the same manner.
- Note that in calculating projectile problems, vertical acceleration must be negative. Also, if a projectile is initially moving upward,
*v*must be positive. If the projectile is initially moving downward,_{i}*v*must be negative. If the projectile is initially at rest (not moving), then_{i}*v*must be 0._{i} - When a projectile is initially moving upward, such as a ball tossed underhanded into the air, the downward acceleration of gravity causes the velocity to reduce at a constant rate. At the highest point in the projectile's arc, its vertical velocity becomes 0 for an instant before becoming negative as the ball begins to move downward.

- Horizontal Motion
- Assuming negligible air resistance, the horizontal velocity of a projectile remains constant throughout its entire flight,
*v*=_{x,i}*v*._{x,f} - Since the horizontal velocity is constant,
*a*= 0. So by substituting this into the third Big 5 equation, we can eliminate the first term, leaving us with*x*=*v*+_{x,i}t*x*. So, the displacement of a projectile after a given amount of time_{i}*t*is*Δx*=*v*._{x,i}t

- Assuming negligible air resistance, the horizontal velocity of a projectile remains constant throughout its entire flight,

- Projectiles Lauched at an angle
- Most projectile problems on the AP exam deal with projectiles that are launched from the ground with an initial velocity that is at an angle θ to the ground. In physics, θ is always given in degrees.
- This initial velocity cannot be used directly in the formulas for the projectile's vertical and horizontal motion because it is not solely horizontal or vertical. Instead, it must be broken into its horizontal and vertical, or
*x*and*y*components. - Components are vectors that point only along the axes of the coordinate system, but together give the same result as the original vector. Putting a vector that is at an angle to the axis together with it's components always forms a right triangle, meaning we can use basic trigonometry and the Pythagorean Theorem when dealing with components.
- When θ and
*v*are given, we use sine and cosine to find the components,_{i}*v*and_{x,i}*v*. See the sample problems for more details on this._{y,i} - Once
*v*and_{x,i}*v*are known, we use these values in place of_{y,i}*v*in their respective equations and solve as usual._{i}