Position, Velocity, and Acceleration. Mr. Miehl
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1 Position, Velocity, and Acceleration Mr. Miehl
2 Velocity is the rate of change of position with respect to time. Velocity ΔD = Δ T Acceleration is the rate of change of velocity with respect to time. ΔV Acceleration = Δ T
3 Warning: Professional driver, do not attempt! When you re driving your car
4 squeeeeek! and you jam on the brakes
5 and you feel the car slowing down
6 what you are really feeling
7 is actually acceleration.
8 I felt that acceleration.
9 How do you find a function that describes a physical event? Steps for Modeling Physical Data 1) Perform an experiment. 2) Collect and graph data. 3) Decide what type of curve fits the data. 4) Use statistics to determine the equation of the curve.
10 A crab is crawling along the edge of your desk. Its location (in feet) at time t (in seconds) is given by P (t ) = t 2 + t. a) Where is the crab after 2 seconds? b) How fast is it moving at that instant (2 seconds)?
11 A crab is crawling along the edge of your desk. Its location (in feet) at time t (in seconds) is given by P (t ) = t 2 + t. a) Where is the crab after 2 seconds? P 2 = ( ) ( ) ( ) 2 P ( 2) = 6 feet
12 A crab is crawling along the edge of your desk. Its location (in feet) at time t (in seconds) is given by P (t ) = t 2 + t. b) How fast is it moving at that instant (2 seconds)? () 2 Pt = t + t V t Velocity function Velocity is the rate of change of position. = P' t = () ( ) 2t + 1 P'2 ( ) = 22 ( ) + 1 P'2 ( ) = 5 feet per second
13 A disgruntled calculus student hurls his calculus book in the air.
14 The position of the calculus book: pt = 16t + 96t ( ) 2 t is in seconds and p(t) is in feet a) What is the maximum height attained by the book? b) At what time does the book hit the ground? c) How fast is the book moving when it hits the ground?
15 a) What is the maximum height attained by the book? The book attains its maximum height when its velocity is 0. Velocity function pt = 16t + 96t ( ) 2 vt ( ) = ( ) p t = 32t = 32t t = 96 t = 3 seconds p 3 = ( ) ( ) ( ) p ( 3) = p ( 3) = 144 feet 2
16 b) At what time does the book hit the ground? The book hits the ground when its position is 0. pt = 16t + 96t ( ) 2 2 0= 16t = 16 tt ( 6) 16t = 0 t 6= 0 t = 0 sec. t = 6 t sec.
17 c) How fast is the book moving when it hits the ground? Good guess: 0 ft/sec This is incorrect. vt ( ) = 32t+ 96 v ( 6) = 32( 6) + 96 v ( 6) = v ( 6) = 96 ft/sec Downward direction
18 Acceleration: the rate of change of velocity with respect to time. Velocity function Acceleration function vt ( ) = 32t+ 96 at = = ft/sec 2 ( ) v( t) 32 How is the acceleration function related to the position function? Acceleration is the second derivative of position. at = p t ( ) ( ) Jerk is the rate of change of acceleration with respect to time.
19 A red sports car is traveling, and its position P (in miles) at time t (in hours) is given by P (t ) = t 2 7t. a) When is the car 30 miles from where it started? b) What is the velocity at the very moment the car is 30 miles away? c) What is the acceleration at the very moment the car is 30 miles away? d) When does the car stop?
20 A red sports car is traveling, and its position P (in miles) at time t (in hours) is given by P (t ) = t 2 7t. a) When is the car 30 miles from where it started? 2 30 = t 7 2 0= t 7t 30 0= t 10 t+ 3 t ( )( ) t 10 = 0 t + 3= 0 t =10 hours t = 3
21 A red sports car is traveling, and its position P (in miles) at time t (in hours) is given by P (t ) = t 2 7t. b) What is the velocity at the very moment the car is 30 miles away? V t = P' t = 2t 7 () () V t = P' t = 2t 7 () () P'10 ( ) = 210 ( ) 7 P'10 ( ) = 13 Miles per hour
22 A red sports car is traveling, and its position P (in miles) at time t (in hours) is given by P (t ) = t 2 7t. c) What is the acceleration at the very moment the car is 30 miles away? V t = P' t = 2t 7 () () At = P'' t = 2 Miles per hour 2 () ( )
23 A red sports car is traveling, and its position P (in miles) at time t (in hours) is given by P (t ) = t 2 7t. d) When does the car stop? V t = P' t = 2t 7 () () 0= 2t 7 7= 2t t = 3.5 hours
24 Conclusion The velocity function is found by taking the derivative of the position function. In order for an object traveling upward to obtain maximum position, its instantaneous velocity must equal 0. As an object hits the ground, its velocity is not 0, its height is 0. The acceleration function is found by taking the derivative of the velocity function.
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