Mechanics: Motion in One Dimension x dx Notes by: Ted Vittitoe dt x t Notes by: Ted Vittitoe Kyle Curry Kinematics in One Dimension (Phy 2053) vittitoe

Motion in One Dimension Sections 2-01 Displacement 2-02 Velocity 2-03 Acceleration 2-04 Motion Diagrams 2-05 One Dimensional Motion with Constant Acceleration 2-06 Freely Falling Objects Kinematics in One Dimension (Phy 2053) vittitoe

An inside observer does not feel he/she is moving. Frame of Reference A measure of distance, speed, and acceleration are always made in some type of Frame of Reference. An inside observer does not feel he/she is moving. An outside observer sees the person inside the train moving at the same velocity as the train Kinematics in One Dimension (Phy 2053) vittitoe

Vectors vs. Scalars A VECTOR QUANTITY requires both a direction and a magnitude to describe. A SCALAR QUANTITY is described only by magnitude. Example: Velocity (a vector) of a track runner may be described as “13 m/s toward the finish line” while her speed (a scalar) would be described as just “13 m/s.” Vector Scalar Velocity Speed Acceleration Energy (Potential, Kinetic) Displacement Mass, Density Forces, Fields Time, Volume Torque, Momentum Temperature

A particle is a point-like mass having infinitesimal Displacement v  In the study of kinematics (how do objects move), we consider a moving object as a particle. A particle is a point-like mass having infinitesimal size and a finite mass. Kinematics in One Dimension (Phy 2053) vittitoe

Displacement The displacement of a particle is defined as its change in position (where the object is currently located). x (m) -6 -4 -2 2 4 6 Dx = x - xo = 6 m - 2 m = 4 m x = Where the object is on a coordinate plane xo = Where the object is Note: Displacement to the right is positive Kinematics in One Dimension (Phy 2053) vittitoe

= -6 m - 6 m Displacement The displacement of a particle is defined as its change in position. x (m) -6 -4 -2 2 4 6 Dx = x - xo = -6 m - 6 m = -12 m Note: Displacement to the left is negative Kinematics in One Dimension (Phy 2053) vittitoe

= (2 m) - (-6 m) Displacement The displacement of a particle is defined as its change in position. x (m) -6 -4 -2 2 4 6 Dx = x - xo = (2 m) - (-6 m) = 8 m Note: Displacement to the right is positive Kinematics in One Dimension (Phy 2053) vittitoe

A student walks 70 m East, then walks 30 m West. Displacement A student walks 70 m East, then walks 30 m West. What is the magnitude of the students net displacement? A) 30 m B) 40 m C) 70 m D) 100 m East West 70 m 30 m Displacement 40 m Kinematics in One Dimension (Phy 2053) vittitoe

Velocity is represented displacement-time graph Average velocity: Total Displacement ÷ Total Time Interval The average velocity of a particle is defined as x x1 x2 t1 t2 Dx Dt Velocity is represented by the slope on a displacement-time graph t Kinematics in One Dimension (Phy 2053) vittitoe

Always a positive value (Magnitude of Average Velocity) Average speed: Always a positive value (Magnitude of Average Velocity) The average speed of a particle is defined as Kinematics in One Dimension (Phy 2053) vittitoe

Instantaneous velocity displacement-time graph The instantaneous velocity v, equals the limiting value of the ratio x t Dx Dt Instantaneous velocity is represented by the slope of a displacement-time graph Kinematics in One Dimension (Phy 2053) vittitoe

The instantaneous speed of a particle is defined Velocity Instantaneous speed The instantaneous speed of a particle is defined as the magnitude of its instantaneous velocity. Kinematics in One Dimension (Phy 2053) vittitoe

Acceleration is represented Average acceleration The average acceleration of a particle is defined as the change in velocity Dvx divided by the time interval Dt during which that change occurred. v v1 v2 t1 t2 Dv Dt Acceleration is represented by the slope on a velocity-time graph t Kinematics in One Dimension (Phy 2053) vittitoe

A new car manufacturer advertises that their car can Acceleration A new car manufacturer advertises that their car can go "from zero to sixty in 8 s". This is a description of A) instantaneous acceleration. B) average speed. C) instantaneous speed. D) average acceleration. Kinematics in One Dimension (Phy 2053) vittitoe

Instantaneous acceleration The instantaneous acceleration equals the derivative of the velocity with respect to time v t Dv Dt Instantaneous acceleration is represented by the slope of a velocity-time graph Kinematics in One Dimension (Phy 2053) vittitoe

Acceleration: Deceleration When an object is slowing down. This does NOT mean that acceleration is negative V0 = 15.0 m/s V = 5.0 m/s t0 = 0.0 s t = 5.0 s a Negative X Direction

Acceleration: Acceleration (Negative Direction) V = - 15.0 m/s V0 = - 5.0 m/s t = 5.0 s t0 = 0.0 s a Negative X Direction Direction

Acceleration: Acceleration Positive Direction X Direction V = - 5.0 m/s V0 = - 15.0 m/s t = 5.0 s t0 = 0.0 s a Positive Direction X Direction

A) traveling at 1.5 m/s in every second. Acceleration A moving car experiences a constant acceleration of 1.5 m/s2. This means the car is A) traveling at 1.5 m/s in every second. B) changing its velocity by 1.5 m/s. C) increasing its velocity by 1.5 m/s in every second. D) increases its displacement by 1.50 m each second. Kinematics in One Dimension (Phy 2053) vittitoe

(a) A car must always have an acceleration in the same Quick Quiz 2.2 (page 32) True or False? (a) A car must always have an acceleration in the same direction as its velocity False (b) It’s possible for a slowing car to have a positive acceleration True (c) An object with constant nonzero acceleration can never stop and stay stopped. True Kinematics in One Dimension (Phy 2053) vittitoe

Formulas for Motion in One Dimension These are the formulas that we will be using to solve problems of 1-D Motion in this course and on the AP Physics B Exam.

Motion Diagrams The displacement versus time for a certain particle moving along the x axis is shown below. Find the average velocity in the time intervals (a) 0 to 2 s (b) 0 to 4 s (c) 2 s to 4 s (d) 4 s to 7 s (e) 0 to 8 s. Kinematics in One Dimension (Phy 2053) vittitoe

Motion Diagrams (con’t) The displacement versus time for a certain particle moving along the x axis is shown below. Find the average velocity in the time intervals (a) 0 to 2 s (b) 0 to 4 s (c) 2 s to 4 s (d) 4 s to 7 s (e) 0 to 8 s. Kinematics in One Dimension (Phy 2053) vittitoe

Motion Diagrams (con’t) The displacement versus time for a certain particle moving along the x axis is shown below. Find the average velocity in the time intervals (a) 0 to 2 s (b) 0 to 4 s (c) 2 s to 4 s (d) 4 s to 7 s (e) 0 to 8 s. Kinematics in One Dimension (Phy 2053) vittitoe

Motion Diagrams (con’t) The displacement versus time for a certain particle moving along the x axis is shown below. Find the average velocity in the time intervals (a) 0 to 2 s (b) 0 to 4 s (c) 2 s to 4 s (d) 4 s to 7 s (e) 0 to 8 s. Kinematics in One Dimension (Phy 2053) vittitoe

Motion Diagrams (con’t) The displacement versus time for a certain particle moving along the x axis is shown below. Find the average velocity in the time intervals (a) 0 to 2 s (b) 0 to 4 s (c) 2 s to 4 s (d) 4 s to 7 s (e) 0 to 8 s. Kinematics in One Dimension (Phy 2053) vittitoe

An object starts from rest and moves with constant acceleration. Motion Diagrams An object starts from rest and moves with constant acceleration. 4 8 12 16 20 24 28 (s) x 4 8 12 16 20 24 28 (m) 1 2 3 t 5 Kinematics in One Dimension (Phy 2053) vittitoe

(s) x 4 8 12 16 20 24 28 (m) 1 2 3 t 5 Motion Diagrams 1 2 3 4 5 t (s) 10 v (m/s) Displacement 25 m Kinematics in One Dimension (Phy 2053) vittitoe

Displacement, velocity and acceleration graphs x Motion Diagrams Displacement, velocity and acceleration graphs x The slope of a displacement-time graph represents velocity t v The slope of a velocity-time graph represents acceleration t a t Kinematics in One Dimension (Phy 2053) vittitoe

Displacement, velocity and acceleration graphs x Motion Diagrams Displacement, velocity and acceleration graphs x The area under a velocity-time graph represents displacement. Dx t v The area under an acceleration-time graph represents change in velocity. Dv t a Dt t Kinematics in One Dimension (Phy 2053) vittitoe

The slope of a position versus time graph gives A) position. Motion Diagrams The slope of a position versus time graph gives A) position. B) velocity. C) acceleration. D) displacement. Kinematics in One Dimension (Phy 2053) vittitoe

The slope of a velocity versus time graph gives A) position. Motion Diagrams The slope of a velocity versus time graph gives A) position. B) velocity C) acceleration D) displacement Kinematics in One Dimension (Phy 2053) vittitoe

One Dimensional Motion with Constant Acceleration Definitions of velocity and acceleration Average velocity Average acceleration Kinematics in One Dimension (Phy 2053) vittitoe

One Dimensional Motion with Constant Acceleration For constant acceleration An object moving with an initial velocity vo undergoes a constant acceleration a for a time t. Find the final velocity. vo ? a time = 0 time = t Solution: Eq 1 Kinematics in One Dimension (Phy 2053) vittitoe

One Dimensional Motion with Constant Acceleration What are we calculating? t a DV Kinematics in One Dimension (Phy 2053) vittitoe

One Dimensional Motion with Constant Acceleration Objects A and B both start at rest. They both accelerate at the same rate. However, object B accelerates for twice the time as object A. What is the final speed of object B compared to that of object A? A) the same speed B) twice as fast C) three times as fast D) four times as fast Kinematics in One Dimension (Phy 2053) vittitoe

One Dimensional Motion with Constant Acceleration For constant acceleration An object moving with a velocity vo is passing position xo when it undergoes a constant acceleration a for a time t. Find the object’s displacement. time = 0 time = t xo ? a vo Solution: Eq 2 Kinematics in One Dimension (Phy 2053) vittitoe

One Dimensional Motion with Constant Acceleration What are we calculating? t vo v at Kinematics in One Dimension (Phy 2053) vittitoe

One Dimensional Motion with Constant Acceleration Objects A and B both start at rest. They both accelerate at the same rate. However, object B accelerates for twice the time as object A. What is the distance traveled by object B compared to that of object A? A) the same distance B) twice as far C) three times as far D) four times as far Kinematics in One Dimension (Phy 2053) vittitoe

One Dimensional Motion with Constant Acceleration Eq 2 Eq 1 Solve Eq 1 for a and sub into Eq 2: Eq 3 Solve Eq 1 for t and sub into Eq 2: Eq 4 Kinematics in One Dimension (Phy 2053) vittitoe

One Dimensional Motion with Constant Acceleration When the velocity of an object is zero, must its acceleration also be zero? A) no, an object thrown upward will have zero velocity at its highest point. B) no, a falling object will have zero velocity after hitting the ground. C) yes, if the object is not moving it can not be accelerating. D) yes, acceleration implies a changing velocity, it can not be zero. Kinematics in One Dimension (Phy 2053) vittitoe

Freely Falling Objects When an object is released from rest and falls in the absence of air resistance, which of the following is true concerning its motion? A) Its acceleration is constant B) Its velocity is constant. C) Neither its acceleration nor its velocity is constant. D) Both its acceleration and its velocity are constant. Kinematics in One Dimension (Phy 2053) vittitoe

(a) How long does it take for the lead car to stop? Problem Two cars are traveling along a straight line in the same direction, the lead car at 25.0 m/s and the other car at 30.0 m/s. At the moment the cars are 40.0 m apart, the lead driver applies the brakes, causing his car to have an acceleration of –2.00 m/s2. (a) How long does it take for the lead car to stop? Kinematics in One Dimension (Phy 2053) vittitoe

(b) How far does the lead car travel during the acceleration? Problem Two cars are traveling along a straight line in the same direction, the lead car at 25.0 m/s and the other car at 30.0 m/s. At the moment the cars are 40.0 m apart, the lead driver applies the brakes, causing his car to have an acceleration of –2.00 m/s2. (b) How far does the lead car travel during the acceleration? Kinematics in One Dimension (Phy 2053) vittitoe

Problem Alternate Solutions Kinematics in One Dimension (Phy 2053) vittitoe

Problem Two cars are traveling along a straight line in the same direction, the lead car at 25.0 m/s and the other car at 30.0 m/s. At the moment the cars are 40.0 m apart, the lead driver applies the brakes, causing his car to have an acceleration of –2.00 m/s2. (c) Assuming that the chasing car brakes at the same time as the lead car, what must be the chasing car’s minimum negative acceleration so as not to hit the lead car? Kinematics in One Dimension (Phy 2053) vittitoe

(d) How long does it take for the chasing car to stop? Problem Two cars are traveling along a straight line in the same direction, the lead car at 25.0 m/s and the other car at 30.0 m/s. At the moment the cars are 40.0 m apart, the lead driver applies the brakes, causing his car to have an acceleration of –2.00 m/s2. (d) How long does it take for the chasing car to stop? Kinematics in One Dimension (Phy 2053) vittitoe

Problem Alternate Solutions Kinematics in One Dimension (Phy 2053) vittitoe

A Cessna aircraft has a lift-off speed of 120 km/h. Problem A Cessna aircraft has a lift-off speed of 120 km/h. (a) What minimum constant acceleration does the aircraft require if it is to be airborne after a takeoff run of 240 m? Kinematics in One Dimension (Phy 2053) vittitoe

A Cessna aircraft has a lift-off speed of 120 km/h. Problem A Cessna aircraft has a lift-off speed of 120 km/h. (b) How long does it take the aircraft to become airborne? Kinematics in One Dimension (Phy 2053) vittitoe

(a) How long did it take the race car to travel this distance? Problem A drag racer starts her car from rest and accelerates at 10.0 m/s2 for a distance of 400 m (1/4 mile). (a) How long did it take the race car to travel this distance? Kinematics in One Dimension (Phy 2053) vittitoe

(b) What is the speed of the race car at the end of the run? Problem A drag racer starts her car from rest and accelerates at 10.0 m/s2 for a distance of 400 m (1/4 mile). (b) What is the speed of the race car at the end of the run? Kinematics in One Dimension (Phy 2053) vittitoe

Problem A ball is thrown vertically upward with a speed of 25.0 m/s. (a) How high does it rise? Kinematics in One Dimension (Phy 2053) vittitoe

A ball is thrown vertically upward with a speed of 25.0 m/s. Problem A ball is thrown vertically upward with a speed of 25.0 m/s. (b) How long does it take to reach its highest point? Kinematics in One Dimension (Phy 2053) vittitoe

A ball is thrown vertically upward with a speed of 25.0 m/s. Problem A ball is thrown vertically upward with a speed of 25.0 m/s. (c) How long does the ball take to hit the ground after it reaches its highest point? Kinematics in One Dimension (Phy 2053) vittitoe

A ball is thrown vertically upward with a speed of 25.0 m/s. Problem A ball is thrown vertically upward with a speed of 25.0 m/s. (d) What is its velocity when it returns to the level from which it started? Kinematics in One Dimension (Phy 2053) vittitoe

Kinematics with Constant Acceleration Review Definitions Average velocity Average acceleration Kinematics with Constant Acceleration Kinematics in One Dimension (Phy 2053) vittitoe

Review t x v a t x v a Dt Dv Dx Kinematics in One Dimension (Phy 2053) vittitoe

Problem Solving Skills Review Problem Solving Skills 1. Read the problem carefully 2. Sketch the problem 3. Visualize the physical situation 4. Identify the known and unknown quantities 5. Identify appropriate equations 6. Solve the equations 7. Check your answers Kinematics in One Dimension (Phy 2053) vittitoe

END

Constant Velocity Suppose that an object is moving with a constant velocity. Make a statement concerning its acceleration. A) The acceleration must be constantly increasing. B) The acceleration must be constantly decreasing. C) The acceleration must be a constant non-zero value. D) The acceleration must be equal to zero.

Freely Falling Objects Can an object have increasing speed while the magnitude of its acceleration is decreasing? Support your answer with an example. A) No, this is impossible because of the way in which acceleration is defined. B) No, because if acceleration is decreasing the object will be slowing down. C) Yes, and an example would be an object falling in the absence of air friction. D) Yes, and an example would be an object released from rest in the presence of air friction.

Freely Falling Objects Suppose a ball is thrown straight up. Make a statement about the velocity and the acceleration when the ball reaches the highest point. A) Both its velocity and its acceleration are zero. B) Its velocity is zero and its acceleration is not zero. C) Its velocity is not zero and its acceleration is zero. D) Neither its velocity nor its acceleration is zero.

Freely Falling Objects Ball A is dropped from the top of a building. One second later, ball B is dropped from the same building. As time progresses, the distance between them A) increases. B) remains constant. C) decreases. D) cannot be determined from the information given.