Kinematics Notes Motion in 1 Dimension Physics C 1-D Motion 2007-2008 Bertrand

Average Speed and Average Velocity Physics C 1-D Motion 2007-2008 Average Speed and Average Velocity Average speed describes how fast a particle is moving. It is calculated by: Average velocity describes how fast the displacement is changing with respect to time: always positive sign gives direction Bertrand

Physics C 1-D Motion 2007-2008 Average Acceleration Average acceleration describes how fast the velocity is changing with respect to time. The equation is: sign determines direction Bertrand

Physics C 1-D Motion 2007-2008 Sample problem: A motorist drives north at 20 m/s for 20 km and then continues north at 30 m/s for another 20 km. What is his average velocity? Bertrand

Physics C 1-D Motion 2007-2008 Sample problem: It takes the motorist one minute to change his speed from 20 m/s to 30 m/s. What is his average acceleration? Bertrand

Average Velocity from a Graph Physics C 1-D Motion 2007-2008 Average Velocity from a Graph A B x Dx Dt t Bertrand

Average Velocity from a Graph Physics C 1-D Motion 2007-2008 Average Velocity from a Graph A B t x Dx Dt Bertrand

Average Acceleration from a Graph Physics C 1-D Motion 2007-2008 Average Acceleration from a Graph A B v Dv Dt t Bertrand

Sample problem: From the graph, determine the average velocity for the particle as it moves from point A to point B. Physics C 1-D Motion 2007-2008 -1 -2 1 2 0.1 0.2 0.3 0.4 0.5 -3 3 t(s) x(m) A B Bertrand

Sample problem: From the graph, determine the average speed for the particle as it moves from point A to point B. Physics C 1-D Motion 2007-2008 -1 -2 1 2 0.1 0.2 0.3 0.4 0.5 -3 3 t(s) x(m) A B Bertrand

Physics C 1-D Motion Instantaneous Speed, Velocity, and Acceleration 2007-2008 Instantaneous Speed, Velocity, and Acceleration Bertrand

Average Velocity from a Graph Physics C 1-D Motion 2007-2008 Average Velocity from a Graph A t x B Remember that the average velocity between the time at A and the time at B is the slope of the connecting line. Bertrand

Average Velocity from a Graph Physics C 1-D Motion 2007-2008 Average Velocity from a Graph A t x B What happens if A and B become closer to each other? Bertrand

Average Velocity from a Graph Physics C 1-D Motion 2007-2008 Average Velocity from a Graph t x A B What happens if A and B become closer to each other? Bertrand

Average Velocity from a Graph Physics C 1-D Motion 2007-2008 Average Velocity from a Graph t x B A What happens if A and B become closer to each other? Bertrand

Average Velocity from a Graph Physics C 1-D Motion 2007-2008 Average Velocity from a Graph t x B A What happens if A and B become closer to each other? Bertrand

Average Velocity from a Graph Physics C 1-D Motion 2007-2008 Average Velocity from a Graph t x A and B are effectively the same point. The time difference is effectively zero. B A The line “connecting” A and B is a tangent line to the curve. The velocity at that instant of time is represented by the slope of this tangent line. Bertrand

Sample problem: From the graph, determine the instantaneous speed and instantaneous velocity for the particle at point B. Physics C 1-D Motion 2007-2008 -1 -2 1 2 0.1 0.2 0.3 0.4 0.5 -3 3 t(s) x(m) A B Bertrand

Average and Instantaneous Acceleration Physics C 1-D Motion 2007-2008 Average and Instantaneous Acceleration t v Instantaneous acceleration is represented by the slope of a tangent to the curve on a v/t graph. A Average acceleration is represented by the slope of a line connecting two points on a v/t graph. C B Bertrand

Average and Instantaneous Acceleration Physics C 1-D Motion 2007-2008 Average and Instantaneous Acceleration Instantaneous acceleration is zero where slope is constant x Instantaneous acceleration is positive where curve is concave up Instantaneous acceleration is negative where curve is concave down t Bertrand

Physics C 1-D Motion 2007-2008 Sample problem: Consider an object that is dropped from rest and reaches terminal velocity during its fall. What would the v vs t graph look like? t v Bertrand

Physics C 1-D Motion 2007-2008 Sample problem: Consider an object that is dropped from rest and reaches terminal velocity during its fall. What would the x vs t graph look like? t x Bertrand

Estimate the net change in velocity from 0 s to 4.0 s Physics C 1-D Motion 2007-2008 Estimate the net change in velocity from 0 s to 4.0 s a (m/s2) 1.0 t (s) 2.0 4.0 -1.0 Bertrand

Estimate the net displacement from 0 s to 4.0 s Physics C 1-D Motion 2007-2008 Estimate the net displacement from 0 s to 4.0 s v (m/s) 2.0 t (s) 4.0 Bertrand

Physics C 1-D Motion Derivatives 2007-2008 Derivatives Bertrand

Sample problem. From this position-time graph Physics C 1-D Motion Sample problem. From this position-time graph 2007-2008 x t Bertrand

Draw the corresponding velocity-time graph Physics C 1-D Motion Draw the corresponding velocity-time graph 2007-2008 x t Bertrand

Suppose we need instantaneous velocity, but don’t have a graph? Physics C 1-D Motion 2007-2008 Suppose we need instantaneous velocity, but don’t have a graph? Suppose instead, we have a function for the motion of the particle. Suppose the particle follows motion described by something like x = (-4 + 3t) m x = (1.0 + 2.0t – ½ 3 t2) m x = -12t3 We could graph the function and take tangent lines to determine the velocity at various points, or… We can use differential calculus. Bertrand

Instantaneous Velocity Physics C 1-D Motion 2007-2008 Instantaneous Velocity Mathematically, velocity is referred to as the derivative of position with respect to time. Bertrand

Instantaneous Acceleration Physics C 1-D Motion 2007-2008 Instantaneous Acceleration Mathematically, acceleration is referred to as the derivative of velocity with respect to time Bertrand

Instantaneous Acceleration Physics C 1-D Motion 2007-2008 Instantaneous Acceleration Acceleration can also be referred to as the second derivative of position with respect to time. Just don’t let the new notation scare you; think of the d as a baby D, indicating a very tiny change! Bertrand

Evaluating Polynomial Derivatives Physics C 1-D Motion 2007-2008 Evaluating Polynomial Derivatives It’s actually pretty easy to take a derivative of a polynomial function. Let’s consider a general function for position, dependent on time. Bertrand

Physics C 1-D Motion 2007-2008 Sample problem: A particle travels from A to B following the function x(t) = 3.0 – 6t + 3t2. What are the functions for velocity and acceleration as a function of time? What is the instantaneous velocity at 6 seconds? What is the initial velocity? Bertrand

Physics C 1-D Motion 2007-2008 Sample problem: A particle travels from A to B following the function x(t) = 2.0 – 4t + 3t2 – t3. a) What are the functions for velocity and acceleration as a function of time? b) What is the instantaneous acceleration at 6 seconds? Bertrand

Sample problem: A particle follows the function Physics C 1-D Motion 2007-2008 Sample problem: A particle follows the function Find the velocity and acceleration functions. Find the instantaneous velocity and acceleration at 2.0 seconds. Bertrand

Physics C 1-D Motion Kinematic Equation Review 2007-2008 Kinematic Equation Review Bertrand

Here are our old friends, the kinematic equations  Physics C 1-D Motion 2007-2008 Here are our old friends, the kinematic equations  Bertrand

Physics C 1-D Motion 2007-2008 Sample problem (basic): Show how to derive the 1st kinematic equation from the 2nd. Sample problem (advanced): Given a constant acceleration of a, derive the first two kinematic equations. Bertrand

Draw representative graphs for a particle which is stationary. Physics C 1-D Motion 2007-2008 Draw representative graphs for a particle which is stationary. x t Position vs time v t Velocity vs time a t Acceleration vs time Bertrand

Physics C 1-D Motion 2007-2008 Draw representative graphs for a particle which has constant non-zero velocity. x t Position vs time v t Velocity vs time a t Acceleration vs time Bertrand

Physics C 1-D Motion 2007-2008 Draw representative graphs for a particle which has constant non-zero acceleration. x t Position vs time v t Velocity vs time a t Acceleration vs time Bertrand

Physics C 1-D Motion 2007-2008 Sample problem: A body moving with uniform acceleration has a velocity of 12.0 cm/s in the positive x direction when its x coordinate is 3.0 cm. If the x coordinate 2.00 s later is -5.00 cm, what is the magnitude of the acceleration? Bertrand

Physics C 1-D Motion 2007-2008 Sample problem: A jet plane lands with a speed of 100 m/s and can accelerate at a maximum rate of -5.00 m/s2 as it comes to a halt. a) What is the minimum time it needs after it touches down before it comes to a rest? b) Can this plane land at a small tropical island airport where the runway is 0.800 km long? Bertrand

Physics C 1-D Motion Freefall 2007-2008 Freefall Bertrand

Physics C 1-D Motion 2007-2008 Free Fall Free fall is a term we use to indicate that an object is falling under the influence of gravity, with gravity being the only force on the object. Gravity accelerates the object toward the earth the entire time it rises, and the entire time it falls. The acceleration due to gravity near the surface of the earth has a magnitude of 9.8 m/s2. The direction of this acceleration is DOWN. Air resistance is ignored. Bertrand

Physics C 1-D Motion 2007-2008 Sample problem: A student tosses her keys vertically to a friend in a window 4.0 m above. The keys are caught 1.50 seconds later. a) With what initial velocity were the keys tossed? b) What was the velocity of the keys just before they were caught? Bertrand

Physics C 1-D Motion 2007-2008 Sample problem: A ball is thrown directly downward with an initial speed of 8.00 m/s from a height of 30.0 m. How many seconds later does the ball strike the ground? Bertrand