AP Calculus AB: 7.1.2 Solving Word Problems Involving Distance and Velocity
These flashcards guide you through using calculus and mathematical modeling to analyze physical phenomena like motion. They cover constructing models from data, interpreting derivatives to find maximum heights, velocities at specific times, and understanding acceleration as the rate of change of velocity. The content emphasizes applying calculus concepts to real-world scenarios involving distance and velocity.
Solving Word Problems Involving Distance and Velocity
By collecting and analyzing physical data, functions can be found that model physical events.
For an object to attain a maximum position, its instantaneous velocity must equal 0.
Acceleration measures the rate of change of velocity with respect to time. Acceleration is the second derivative of position and the first derivative of velocity.
Key Terms
Solving Word Problems Involving Distance and Velocity
By collecting and analyzing physical data, functions can be found that model physical events.
For an object to attain a maxi...
note 1
Functions are often used to mathematically describe certain events. Using an equation to describe a real world phenomenon is called mathema...
note 2
Finding the velocity of the book when it hits the ground
requires you to know the time the book hit the ground and the velocity equation...
A physicist with too much time on his hands is trying to calculate a position function for Tabby, his pet slug. He sets up a camera to take a picture of Tabby’s initial position and her location at every even minute thereafter.
At t = 0, Tabby is at the starting point.
At t = 2, she is 4 inches away.
At t = 4, she is 8 inches away.
At t = 6, she is 12 inches away.
Which of the following functions is the best fit for Tabby’s position function?
PTabby(t) = 2t
Suppose the altitude of a particular bomb being dropped out of a plane is given by the equation p (t) = −4.9t ^2 + 26t + 10, where t is in seconds and p is in meters. What is the initial velocity of the falling bomb?
26 m / sec
Given that a particular moving object’s position is given by the equation p(t)=−32t^2+110, what is the equation for the object’s acceleration?
a (t) = −64
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Solving Word Problems Involving Distance and Velocity |
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A physicist with too much time on his hands is trying to calculate a position function for Tabby, his pet slug. He sets up a camera to take a picture of Tabby’s initial position and her location at every even minute thereafter. At t = 0, Tabby is at the starting point. Which of the following functions is the best fit for Tabby’s position function? | PTabby(t) = 2t |
Suppose the altitude of a particular bomb being dropped out of a plane is given by the equation p (t) = −4.9t ^2 + 26t + 10, where t is in seconds and p is in meters. What is the initial velocity of the falling bomb? | 26 m / sec |
Given that a particular moving object’s position is given by the equation p(t)=−32t^2+110, what is the equation for the object’s acceleration? | a (t) = −64 |
Suppose the position of a particle on a number line is given by p (t) = 3t^ 3 − 3t, where t is in minutes and p is in meters. Which of the following equations represents the acceleration of the object? | a (t) = 18t |
The alien Shaxxxt is flying around the solar system in his new Insight Driven Flying Saucer. The saucer starts off pretty slow, but it picks up speed quickly. The position function for the saucer in miles is P (t) = 2e 2t where t is in minutes. What is Shaxxxt’s velocity when t = 6? | 651019.17 miles per minute |
A man standing on the top of a cliff throws a nickel over the side. The nickel’s altitude is given by the function P(t)=−16t^2+576,where t is in seconds. When will the altitude of the nickel equal zero? | t = 6 |
A cartoon cat lights a bomb and tosses itinto the air. The bomb is set to detonate in8 seconds. The position of the bomb is given by the function P(t)=−9t^2+72t with P(t) in feet and t in seconds.Where is the bomb’s highest point? | 144 feet |
Consider this graph of P ′ (t). At which t-value is P (t) the greatest? | t3 |
Suppose the altitude of a ball shot into the air is given by the equation p (t) = −4.9t ^2 + 49t, where t is measured in seconds and p is measured in meters. What is the velocity of the object when it strikes the ground? | −49 m / sec |
A cartoon cat lights a bomb and tosses it into the air. The bomb is set to detonate in 8 seconds. The altitude of the bomb is given by the function P (t) = −9t 2 + 72t with P (t) in feet and t in seconds. What is the altitude of the bomb when it explodes? | 0 feet |
A man driving a little red sports car is travelling along the highway. His position is given in miles by P (t) = 3000t ^2 + 40t, with t in hours. The speed limit is 70 mph. At what time does the man reach the speed limit? | 18 seconds |
A 3-egg omelette is hurled by a disgruntledcustomer over the side of a building. The distance of the omelette from the customer in feet is given by P(t)=16t^2. t is measured in seconds. The omelette will hit the ground in 5 seconds. With what velocity does the omelette hit the ground? | 160 feet per second |