Flashcards in 7.1.2 Solving Word Problems Involving Distance and Velocity Deck (15):
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.
- Functions are often used to mathematically describe certain events. Using an equation to describe a real world phenomenon is called mathematical modeling.
- Before you can use calculus to study a situation, the situation must be modeled.
- Modeling requires you to construct an experiment and gather data. Once enough data is accumulated you try to fit an equation to the data points to describe them mathematically. Entire courses are devoted to the techniques of modeling.
- Scientific theories and formulas often rely on mathematical modeling.
- The altitude of a book thrown into the air can be modeled by a quadratic equation.
- You can apply calculus to a mathematical model to discover other facts about the event.
- To determine when the object reached its maximum height, you can find the derivative of the position function. When the velocity of the book changes from positive to negative the book changes from rising to falling. It is at this moment that the book is at its maximum height, namely when the slope of the tangent line is equal to 0.
- Set the derivative equal to 0 to find the time that the book reaches its maximum height. Plug that time into the position function to determine the maximum height.
- The book hits the ground when its position (altitude) is equal to 0. To determine the time when this happens, set the position equation equal to 0.
- It is important to consider what different actions mean when working with models. For example, when you set the variable equal to a specific time, what you are actually doing is finding the position of the object at that time. Setting the position function equal to a value tells you the time at which the object was at that altitude.
- 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. Just plug that time into the velocity equation to find the speed.
- Although the book will be traveling 0 ft/sec immediately after the book hits the ground, notice that the question asks for the instantaneous velocity the moment the book hits the ground, not immediately after.
- The rate of change of velocity is called acceleration.
Acceleration describes how the velocity of an object is
changing. Find the acceleration function by taking the
derivative of the velocity function.
- The way that acceleration changes can also be examined. The rate of change of acceleration is called the jerk. The jerk function is the derivative of the acceleration function. The jerk function does not have many practical applications in motion.
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
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?
Consider this graph of P ′ (t).
At which t-value is P (t) the greatest?
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?
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?