KINEMATICS Equations … The BIG FOUR

Competency Goal 2: Build an understanding of linear motion. Objectives – Be able to: 2.03 Analyze acceleration as rate of change in velocity Using graphical and mathematical tools, design and conduct investigations of linear motion and the relationships among:  Position.  Average velocity.  Instantaneous velocity.  Acceleration.  Time.

KINEMATIC EQUATIONS It is now possible to describe the motion of an object traveling with a constant acceleration along a straight path. To do so, we use a set of equations that entail no new concepts, because they are obtained by combining the familiar ideas of displacement, velocity, and acceleration.

KINEMATIC EQUATIONS Whenever possible, it will be convenient to place the frame of reference at the origin x 0 = 0 m when t 0 = 0 s. With this assumption, the displacement Δx = x – x 0 becomes Δx = x.

KINEMATIC EQUATIONS Let’s review the quantities we’ve seen so far: The fundamental quantities are displacement (x or y), velocity (v), and acceleration (a). Acceleration is a change in velocity, from an initial velocity (v 0 or v i ) to a final velocity (v or v f ).

KINEMATIC EQUATIONS And finally, the motion takes place during some elapsed time interval, Δt. If we agree to start our clocks at time t i = 0, then we can just write t instead of Δt, which simplifies the notation. Therefore, we have five kinematic quantities: x, v 0, v, a, and t.

KINEMATIC EQUATIONS These five quantities are related by a group of equations that we call the BIG FOUR: Variable missing BIG FOUR #1: x = ½(v i + v f )t a BIG FOUR #2: v f = v i + at x BIG FOUR #3: x f = x i + v i t + ½at 2 v f BIG FOUR #4: v f 2 = v i 2 + 2ax t

KINEMATIC EQUATIONS MEMORIZE the BIG FOUR in the next two days: Variable missing BIG FOUR #1: x = ½(v i + v f )t a BIG FOUR #2: v f = v i + at x BIG FOUR #3: x f = x i + v i t + ½at 2 v f BIG FOUR #4: v f 2 = v i 2 + 2ax t Make a flash card for each formula – missing variable on the back.

KINEMATICS BIG FOUR In BIG FOUR #1, the average velocity is simply the average of the initial velocity and the final velocity: v = ½(v i + v f ). (This is a consequence of the fact that the acceleration is constant.)

KINEMATICS BIG FOUR Each of the BIG FOUR equations is missing one of the five fundamental quantities. The way you decide which of equation to use when solving a problem is to determine which of the fundamental quantities is missing from the problem – that is, which quantity is neither given nor asked for – and then use the equation that doesn’t have that variable.

KINEMATICS BIG FOUR For example, if the problem never mentions the final velocity … … v is neither given nor asked for … … the equation to use is the one that’s missing v f … …that’s BIG FOUR #3 … x f = x i + v i t + ½at 2

KINEMATICS BIG FOUR BIG FOUR #1 and #2 are simply definitions of v and a written in forms that don’t involve fractions. The other BIG FOUR equations can be derived from these two definitions and the equation v = ½(v i + v f ) using a bit of algebra.

Example The Displacement of a Speedboat The speedboat in the figure has a consant acceleration of +2.0 m/s 2. If the initial velocity of the boat is +6.0 m/s, find its displacement after 8.0 seconds.

Example The Displacement of a Speedboat Reasoning Numerical values for the three unknown variables are listed in the data table. We’re asked to determine the displacement x of the speedboat, so it gets the question mark. We choose BIG FOUR #3, x = x 0 + v 0 t + ½at 2 x = 0 + (6.0 m/s)(8.0 s) + ½(+2.0 m/s 2 )(8.0 s) 2 x = m + 64 m = 110 m. A calculator would give the answer as 112 m, but this number must be rounded to 110 m, since the data are accurate to only two significant digits. xavv0v0 t ?+2.0 m/s m/s8.0 s

Example The Displacement of a Speedboat Alternate Reasoning: We can use a two step method to solve the same problem: If the final velocity can be found, we can use the x = vt = ½(v 0 + v)t if a value for the final velocity v can be found. To find the final velocity, it is necessary to use the value given for the acceleration, because it tells us how the velocity changes, according to BIG FOUR #2, v = v 0 + aΔt. xavv0v0 t ?+2.0 m/s m/s8.0 s

Example The Displacement of a Speedboat Solution The final velocity is v f = v i + aΔt = 6.0 m/s + (2.0 m/s 2 )(8. s) = +22 m/s The displacement of the boat can now be obtained: x = ½ (v 0 + v)t = ½ (6.0 m/s + 22 m/s)(8.0 s) = +110 m A calculator would give the answer as 112 m, but this number must be rounded to 110 m, since the data are accurate to only two significant digits.

Example Catapulting a Jet A jet is taking off from the deck of an aircraft carrier, as the figure shows. Starting from rest, the jet is catapulted with a constant acceleration of +31 m/s 2 along a straight line and reaches a velocity of +62 m/s. Find the displacement of the jet:

Example Catapulting a Jet Reasoning The data are as follows: The initial velocity v 0 is zero, since the jet starts from rest. The displacement x of the aircraft can be obtained from BIG FOUR #1 x = 1/2(v 0 + v)t, if we can determine the time t during which the plane is being accelerated. But t is controlled by the value of the acceleration. With larger accelerations, jet reaches its final velocity in shorter times, as can be seen by solving BIG FOUR #2 v = v 0 + at for t. xavv0v0 t ?+31 m/s m/s0 m/s

Example Catapulting a Jet Solving BIG FOUR #2 for t, v - v 0 62 m/s - 0 m/s t = = = 2.0 s a 31 m/s 2 Since the time is now known, the displacement can be found by using BIG FOUR #1: x = 1/2(v 0 + v)t = 1/2(0 m/s + 62 m/s)(2.0 s) = +62 m

KINEMATIC EQUATIONS MEMORIZE the BIG FOUR in the next two days: Variable missing BIG FOUR #1: x = ½(v i + v f )t a BIG FOUR #2: v f = v i + at x BIG FOUR #3: x f = x i + v i t + ½at 2 v f BIG FOUR #4: v f 2 = v i 2 + 2ax t Make a flash card for each formula – missing variable on the back.

Competency Goal 2: Build an understanding of linear motion. Objectives – Be able to: 2.03 Analyze acceleration as rate of change in velocity Using graphical and mathematical tools, design and conduct investigations of linear motion and the relationships among:  Position.  Average velocity.  Instantaneous velocity.  Acceleration.  Time.