4.2 A Model for Accelerated Motion

Chapter Objectives  Calculate acceleration from the change in speed and the change in time.  Give an example of motion with constant acceleration.  Determine acceleration from the slope of the speed versus time graph.  Calculate time, distance, acceleration, or speed when given three of the four values.  Solve two-step accelerated motion problems.  Calculate height, speed, or time of flight in free fall problems.  Explain how air resistance makes objects of different masses fall with different accelerations.

Chapter Vocabulary  acceleration  acceleration due to gravity (g)  air resistance  constant acceleration  delta (Δ)  free fall  initial speed  m/s 2  term  terminal velocity  time of flight  uniform acceleration

Inv 4.2 Accelerated Motion Investigation Key Question: How does acceleration relate to velocity?

4.2 A Model for Accelerated Motion  To get a formula for solve for the speed of an accelerating object, we can rearrange the experimental formula we had for acceleration.

4.2 The speed of an accelerating object  In physics, a piece of an equation is called a term.  One term of the formula is the object’s starting speed, or its initial velocity (v 0 )  The other term is the amount the velocity changes due to acceleration.

1.You are asked for speed. 2.You are given initial speed, acceleration and time. 3.Use the relationship v = v 0 + at 4.Substitute values  v = 2 m/s + (0.75 m/s 2 )(10 s)  v = 9.5 m/s 2 Calculating speed A ball rolls at 2 m/s off a level surface and down a ramp. The ramp creates an acceleration of 0.75 m/s 2. Calculate the speed of the ball 10 s after it rolls down the ramp.

4.2 Distance traveled in accelerated motion  The distance traveled by an accelerating object can be found by looking at the speed versus time graph.  The graph shows a ball that started with an initial speed of 1 m/s and after one second its speed has increased.

4.2 Distance traveled in accelerated motion  The area of the shaded rectangle is the initial speed v 0 multiplied by the time t, or v 0 t.  The second term is the area of the shaded triangle.

4.2 A Model for Accelerated Motion  It is possible that a moving object may not start at the origin.  Let x 0 be the starting position.  The distance an object moves is equal to its change in position (x – x 0 ).

1.You are asked for distance. 2.You are given initial speed and acceleration. Assume an initial position of 0 and a final speed of 0. 3.Use the relationship v = v 0 + at and x = x 0 + v 0 t + 1/2at 2 4.At the highest point the speed of the ball must be zero. Substitute values to solve for time, then use time to calculate distance.  0 = 2 m/s + (- 0.5 m/s 2 )(t) = - 2 m/s = m/s 2 (t) t = 4 s  x = (0) + (2 m/s) ( 4 s) + (0.5) (-0.5 m/s 2 ) (4 s) 2 = 4 meters Calculating position from speed and acceleration The angle of the ramp creates an acceleration of m/s 2. What distance up the ramp does the ball travel before it turns around and rolls back? A ball traveling at 2 m/s rolls up a ramp.

4.2 Solving motion problems with acceleration  Many practical problems involving accelerated motion have more than one step.  List variables  Cancel terms that are zero.  Speed is zero when it starts from rest.  Speed is zero when it reaches highest point  Use another formula to find the missing piece of information.

1.You are asked to find the length of the ramp. 2.You are given v 0 = 0, v = 2 m/s at t = 1 s, t = 3 s at the bottom of the ramp, and you may assume x 0 = 0. 3.After canceling terms with zeros, v = at and x = ½ at 2 4.This is a two-step problem. First, calculate acceleration, then you can use the position formula to find the length of the ramp.  a = v ÷ t = (2 m/s ) ÷ (1 s ) = 2 m/s 2  x = ½ at 2 = (0.5)(2 m/s )(3 s ) 2 = 9 meters Calculating position from time and speed After one second, the speed of the ball is 2 m/s. How long does the ramp need to be so that the ball can roll for 3 seconds before reaching the end? A ball starts to roll down a ramp with zero initial speed.

1.You are asked to find the time and speed. 2.You are given v 0 = 0, x = 440 m, and a = 6 m/s 2 ; assume x 0 = 0. 3.Use v = v 0 + at and x = x 0 + v 0 t + ½ at 2 4.Since x 0 and v 0 = 0, the equation reduces to x = ½at 2  440 m = (0.5)(6 m/s 2 ) (t) 2  t 2 = 440 ÷ 3 = s t = 12.1 s Calculating time from distance and acceleration A car at rest accelerates at 6 m/s 2. How long does it take to travel 440 meters, or about a quarter-mile, and how fast is the car going at the end?