Motion Problems (you may use a calculator for these problems)

Transcription

1 AP Calculus CHAPTER WORKSHEET DERIVATIVES Name Seat # Motion Problems (you may use a calculator for these problems) Date. A particle moves along a line so that its position at any time t 0 is given by the function s t t t, where s is measured in meters and t is measured in seconds. a) Find the position after the first 5 seconds. b) Find the average velocity during the first 5 seconds. Indicate units of measure. c) Find the instantaneous velocity when t = 4. Indicate units of measure. d) Find the acceleration of the particle when t = 4. Indicate units of measure. e) Was the particle speeding up or slowing down at t = 4? Explain. f) At what values of t does the particle change direction?. A rock thrown vertically upward from the surface of the moon at velocity of 4 m/sec reaches a height of s 4t 0.8t meters in t seconds. a) Find the rock s velocity and acceleration as functions of time. (The acceleration in this problem is the value of gravity on the moon!) b) How long did it take for the rock to reach its highest point? (Hint: what is he velocity at the highest point?) c) How high did the rock go? d) When did the rock reach half its maximum height? e) How long was the rock aloft?. The equation for free fall near the surface of Mars (s in meters, t in seconds) is, s.86t long would it take for a rock falling from rest to reach a velocity of 6.6 m/sec on Mars?. How 4. A bullet fired straight up from the Earth s surface would reach a height of t seconds. How long would it take the bullet to get back down? s 8t 6t ft after 5. The position of a body at time t sec is s t 6t 9t meters. Find the body s acceleration each time the velocity is zero. Indicate units of measure. 6. A body s velocity at time t min is v t 9t t 5 ft/min. a) Find the body s velocity each time the acceleration is zero. Indicate units of measure. b) Find the body s speed each time the acceleration is zero. Indicate units of measure. SEE OTHER SIDE

2 7. The figure shows the velocity t ds dt v in m/sec of a body moving along a coordinate line. The x-axis shows time in seconds. a) When does the body reverse directions? b) When is the body moving at a constant speed? c) Graph the body s speed for 0 t 0. d) Find the acceleration at t = and t = 9 sec. Indicate units of measure. e) Graph the body s acceleration, where defined. 8. The figure shows the position t s in yard of a particle moving along a number line. The x-axis shows time in minutes. a) Find the velocity of the particle at t = 0.5 and t =.5 minutes. Indicate units of measure. b) When is the particle moving to the left? Moving to the right? Standing still? c) Graph the particle s velocity (where defined.) d) Graph the particle s speed (where defined.) 9. When a model rocket is launched, the propellant burns for a few seconds, accelerating the rocket upward. After burnout, the rocket coasts upward for a while and then begins to fall. A small explosive charge pops out a parachute shortly after the rocket starts downward. The parachute slows down the rocket to keep it from breaking when it lands. The graph to the right shows velocity data in ft/sec for the flight in the y-axis. The x-axis shows time after launch in seconds. Use the graph to answer the following. a) How fast was the rocket climbing when the engine stopped? b) For how many seconds did the engine burn? c) When did the rocket reach its highest point? What was its velocity then? d) At t = 4 sec, was the rocket speeding up or slowing down? Why? e) When did the parachute pop out? How fast was the rocket falling then? f) How long did the rocket fall before the parachute opened? g) When was the rocket acceleration s greatest? When was the acceleration constant?

3 AP Calculus CHAPTER WORKSHEET DERIVATIVES Motion Problems ANSWER KEY. a) 5 s 5 s 0 b) s m 5 0 m/sec v t s' t t v 4 m/sec a t v' t s" t a 4 m/sec v 4 and a(4) 0 particle was speeding up, velocity and acceleration had the same sign v t t 0 t. sec c) 5 d) e) 0 f) 5 + v(t) 0.5 Changes directions at t =.5 sec. a) t s' t 4.6t a t v' t. 6 b) v t 0 t 5 sec c) s 5 80 m v m/sec d) 90 t 4t 08. t t 4. 9 e) s t 0 t 0 or t = 0 sec s sec and t = sec. t.7t 6.6 t v sec 4. t 0 t 0 s or t = 5 sec 5. v t t t 9 0 t sec or t = sec a t 6t a 6 m/sec and 6 a m/sec 6. a) t 6t 8t 0 t v 0 ft/min and a min or t = min v ft/min b) speed at t = min is 0 ft/min and at t = min is ft/min

4 7. a) At t = sec and t = 7 sec b) < t < 6 c) a ( ) m/sec and a ( 9). 5 m/sec Speed vs time d) Acceleration vs time e)

5 8. a) v ( 0.5) yards/min and v (.5) yards/min b) Moving to the left: < t < and t > 5 min Moving to the right: 0 < t < min Standing still: < t < and < t < 5 min Velocity vs time 5 Speed vs time c) a) 00 ft/sec b) sec c) 8 sec; 0 ft/sec d) v 0 and a 0 the rocket was slowing down at t = 4 seconds e) sec; 00 ft/sec f) sec g) Greatest: sec; constant: < t < sec