8/27/14. Kinematics and One-Dimensional Motion: Non-Constant Acceleration. Average Velocity. Announcements 8.01 W01D3. Δ r. v ave. = Δx Δt.

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1 Kinematics and One-Dimensional Motion: Non-Constant Acceleration 8.01 W01D3 Announcements Familiarize Yourself with Website Buy or Download Textbook (Dourmashkin, Classical Mechanics: MIT 8.01 Course Notes Revised Edition ) Downloadable links (require certificates): courseware/intro/about:resources/ Buy Clicker at MIT Coop Sunday Tutoring in from 1-5 pm Average Velocity The average velocity, v ave (t), is the displacement divided by the time interval Δr v ave Δ r = Δx î = v ave,x (t)î The x-component of the average velocity is given by v ave,x = Δx 1

2 Instantaneous Velocity and Differentiation For each time interval, calculate the x-component of the average velocity v ave,x (t) = Δx / Take limit as 0 sequence of the x-component average velocities Δx lim 0 = lim 0 x(t + ) x(t) dx dt The limiting value of this sequence is x-component of the instantaneous velocity at the time t. (t) = dx / dt Instantaneous Velocity x-component of the velocity is equal to the slope of the tangent line of the graph of x-component of position vs. time at time t (t) = dx dt Worked Example: Differentiation x(t) = At 2 x(t + ) = A(t + ) 2 = At 2 + 2At + A 2 x(t + ) x(t) = 2At + A dx dt = lim x(t + ) x(t) = lim(2at + A) = 2At 0 0 Generalization for Polynomials: x(t) = At n dx n 1 = nat dt 2

3 Concept Question: Instantaneous Velocity The graph shows the position as a function of time for two trains running on parallel tracks. For times greater than t = 0, which of the following is true: 1. At time t B, both trains have the same velocity. 2. Both trains speed up all the time. 3. Both trains have the same velocity at some time before t B,. 4. Somewhere on the graph, both trains have the same acceleration. Average Acceleration Change in instantaneous velocity divided by the time interval = t 2 t 1 a ave Δ v = Δ î = (,2,1 ) î = Δ î = a ave,x î The x-component of the average acceleration a ave,x = Δ Instantaneous Acceleration and Differentiation For each time interval, calculate the x-component of the average acceleration a ave,x (t) = Δ / Take limit as 0 sequence of the x-component average accelerations Δv lim x 0 = lim 0 (t + ) (t) The limiting value of this sequence is x-component of the instantaneous acceleration at the time t. a x (t) = d / dt d dt 3

4 Instantaneous Acceleration The x-component of acceleration is equal to the slope of the tangent line of the graph of the x-component of the velocity vs. time at time t a x (t) = d dt Group Problem: Model Rocket A person launches a home-built model rocket straight up into the air at y = 0 from rest at time t = 0. (The positive y- direction is upwards). The fuel burns out at t = t 0. The position of the rocket is given by y = 1 2 (a g)t 2 a t6 / t 4 0 ; 0 < t < t 0 with a 0 and g are positive. Find the y-components of the velocity and acceleration of the rocket as a function of time. Graph a y vs t for 0 < t < t 0. Non-Constant Acceleration and Integration 4

5 Change in Velocity: Integral of Acceleration Consider some time t such that t 0 < t < t f Then the change in the x-component of the velocity is the integral of the x- component acceleration (denote,0 ) ). (t),0 = a x ( t )d t 0 Integration is the inverse operation of differentiation Change in Position: Integral of Velocity Area under the graph of x-component of the velocity vs. time is the displacement (denote x 0 x )). x(t) x 0 = ( t )d t 0 Worked Example: Time-Dependent Acceleration Acceleration is a non-constant function of time a x (t) = At 2 with t 0 = 0,,0 = 0, and x 0 = 0. Change in velocity: (t) 0 = A t 2 d t = A t 3 v 3 x (t) = At3 t =0 t =0 3 Change in position: x(t) 0 = A t 3 3 d t x(t) = A t 4 12 t =0 = At Generalization for Polynomials: t n d t = n +1 t =t 0 t n+1 t =t0 = t n+1 n +1 t n+1 0 n +1 5

8/27/14 Special Case: Constant Acceleration ax = constant Acceleration: Velocity: vx (t) vx,0

x0 + vx,0t + ax t 2 2 Concept Question: Integration A particle, starting at rest at t = 0,

Differentiating ax(t) twice. 2. Integrating ax(t) twice. 3. (1/2) ax(t) times t2. 4.

Group Problem: Sports Car At t = 0, a sports car starting at rest at x = 0 accelerates with

afterwards with A, B > 0 (1) Find expressions for the velocity and position vectors of the

6 8/27/14 Special Case: Constant Acceleration ax = constant Acceleration: Velocity: vx (t) vx,0 = a x dt = ax t t =0 vx (t) = vx,0 + ax t Position: x(t) x0 = (v x,0 + a x t ) dt 0 1 x(t) = x0 + vx,0t + ax t 2 2 Concept Question: Integration A particle, starting at rest at t = 0, experiences a nonconstant acceleration ax(t). It s change of position can be found by 1. Differentiating ax(t) twice. 2. Integrating ax(t) twice. 3. (1/2) ax(t) times t2. 4. None of the above. 5. Two of the above. Group Problem: Sports Car At t = 0, a sports car starting at rest at x = 0 accelerates with an x-component of acceleration given by ax (t) = At Bt 3, for 0 < t < (A / B)1/2 and zero afterwards with A, B > 0 (1) Find expressions for the velocity and position vectors of the sports car as functions of time for t >0. (2) Sketch graphs of the x-component of the position, velocity and acceleration of the sports car as a function of time for t >0 6

7 Appendix: Integration and the Riemann Sum Mean Value Theorem: For each rectangle there exists a time t i < t = c i < t i+1 such that Change in Velocity: Area Under Curve of Acceleration vs. time (t i+1 ) (t i ) = d dt (c i ) = a x (c i ) Apply Mean Value Theorem (t 1 ) ) = a x (c 0 ) ( (t 2 ) (t 1 )) = a x (c 1 ) = (t i+1 ) (t i ) = a x (c i ) = (t n ) (t n 1 ) = a x (c n 1 ) We can add up the area of the rectangles and find (t n ) ) = i=n 1 i=0 ((a x (c i )) 7

8 Change in Velocity: Integral of Acceleration The area under the graph of the x-component of the acceleration vs. time is the change in velocity (t f ) ) = lim (t f ) ) i 0 i=1 t=t f t=t 0 i=n a x (t)dt a x (t i ) i 8

Kinematics and One Dimensional Motion

Kinematics and One Dimensional Motion Kinematics Vocabulary Kinema means movement Mathematical description of motion Position Time Interval Displacement Velocity; absolute value: speed Acceleration Averages