Kinematics The branch of mechanics that studies the motion of a body without caring about what caused the motion.

Some Physics Quantities Vector - quantity with both magnitude (size) & direction Scalar - quantity with magnitude only Vectors: Displacement Velocity Acceleration Momentum Force Scalars: Distance Speed Time Mass Energy

Scalar Quantities & Vector Quantities Scalar quantities have magnitude Example: speed 15 m/s Vector quantities have magnitude and direction Example: velocity 15 m/s North

Kinematics definitions Kinematics – branch of physics; study of motion Position (x) – where you are located Distance (d ) – how far you have traveled, regardless of direction Displacement (x) – where you are in relation to where you started

Particle Has position and mass. Has NO size or volume. Located at one point in space.

The Particle Model A simplifying model in which we treat the object as if all its mass were concentrated at a single point. This model helps us concentrate on the overall motion of the object.

Position Location of a particle in space. One dimension (x) Two dimensions (x,y) Three dimensions (x,y,z)

1-Dimensional Coordinates x = 1 m -1 1 2 3 X (m)

Distance The total length of the path traveled by an object. Does not depend upon direction. “How far have you walked?”

1-Dimensional Coordinates Distance moved by particle is 2 meters. xi = 1 m xf = -1 m -1 1 2 3 X (m)

Displacement The change in position of an object. Depends only on the initial and final positions, not on path. Includes direction. “How far are you from home?”

Displacement Represented by x. x = xf - xi where xf = final position xi= initial position

1-Dimensional Coordinates Distance moved by particle is 2 meters. Displacement of particle is -2 meters. xi = 1 m xf = -1 m -1 1 2 3 X (m)

Distance vs Displacement B 100 m distance 50 m displacement

Displacement Linear change in position of an object Is not the same as distance

Displacement Distance = length (blue) How many units did the object move? Displacement = change in position (red) How could you calculate the magnitude of line AB? ≈ 5.1 units, NE

Checking Understanding Maria is at position x = 23 m. She then undergoes a displacement ∆x = –50 m. What is her final position? –27 m –50 m 23 m 73 m Answer: A

Answer Maria is at position x = 23 m. She then undergoes a displacement ∆x = –50 m. What is her final position? –27 m –50 m 23 m 73 m Answer: A

Average Speed save = d t save = rate (speed) Where: d = distance t = elapsed time

Average Velocity vave = ∆x ∆t Where: vave = average velocity ∆x = displacement (x2-x1) ∆t = change in time(t2-t1)

Velocity vs Speed Average speed is always positive. Average velocity can be positive or negative depending direction. Absolute value of velocity can be used for speed if the object is not changing direction.

Average Velocity B x Dx A Dt t Vave = Dx/Dt, or the slope of the line connecting A and B.

Average Velocity A x B Dx Dt t Vave = Dx/Dt; still determined by the slope of the line connecting A and B.

Instantaneous Velocity x B Determined by the slope of the tangent to a curve at a single point.

Velocity can be represented graphically: Position Time Graphs

Velocity can be interpreted graphically: Position Time Graphs Find the average velocity between t = 3 min to t = 8 min

Calculate the average velocity for the entire trip

Position-Time Graphs Object at rest? Traveling slowly in a positive direction? Traveling in a negative direction? Traveling quickly in a positive direction?

Acceleration A change in velocity is called acceleration. Acceleration can be speeding up slowing down turning

Uniformly Accelerated Motion In Physics, we will generally assume that acceleration is constant. With this assumption we are free to use this equation: a = ∆v ∆t

Units of Acceleration The SI unit for acceleration is m/s2.

Sign of Acceleration Acceleration can be positive or negative. The sign indicates direction. .

General Rule If the sign of the velocity and the sign of the acceleration is the same, the object speeds up. If the sign of the velocity and the sign of the acceleration are different, the object slows down.

Is an object speeding up or slowing down? Depends upon the signs of both velocity and acceleration Velocity Accel Motion + Speeding up in + dir - Speeding up in - dir Slowing down in + dir Slowing down in - dir

Accelerating objects… Note: each of these curves has many different slopes (many different velocities)! x t

Pick the constant velocity graph(s)… x v A C t t x v B D t t

Summary: Constant position graphs x t Position vs time v t Velocity vs time a t Acceleration vs time

Summary: Constant velocity graphs x t Position vs time v t Velocity vs time a t Acceleration vs time

Summary: Constant acceleration graphs x t Position vs time v t Velocity vs time a t Acceleration vs time

Velocity-Time Graphs Is this object accelerating? How do you know? What can you say about its motion?

Velocity-Time Graph Is this object accelerating? How do you know? What can you say about its motion? What feature of the graph represents acceleration? www.gcsescience.com

Summary v = vo + at x = xo + vot + 1/2 at2 v2 = vo2 + 2a(∆x)

Free Fall Occurs when an object falls unimpeded. Gravity accelerates the object toward the earth.

Acceleration due to gravity g = 9.8 m/s2 downward. a = -g if up is positive. acceleration is down when ball is thrown up EVERYWHERE in the balls flight.

Summary v = vo - gt x = xo + vot - 1/2 gt2 v2 = vo2 – 2g(∆x)

Symmetry When something is thrown upward and returns to the thrower, this is very symmetric. The object spends half its time traveling up; half traveling down. Velocity when it returns to the ground is the opposite of the velocity it was thrown upward with. Acceleration is –9.8 m/s2 everywhere!