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AP Calculus AB: 10.1.3 Solving Vertical Motion Problems

Mathematics8 CardsCreated 3 months ago

This section focuses on solving vertical motion problems using position and velocity functions derived from constant gravitational acceleration. It includes finding distances fallen, initial velocities for specific heights, times when objects are motionless, and analyzing motion scenarios like object crossing—all by applying calculus and kinematic principles.

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Solving Vertical Motion Problems

  • Given the initial velocity and initial position of an object moving vertically, you can use the fact that the acceleration due to gravity is a constant to find the velocity and position functions.

  • The velocity and position functions can be used to answer many questions about the motion of an object.

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Key Terms

Term
Definition

Solving Vertical Motion Problems

  • Given the initial velocity and initial position of an object moving vertically, you can use the fact that the acceleration due to gravity i...

note

  • This problem asks you to use the length of time an object was falling to determine the distance it fell.

  • A falling object ha...

The position of a helicopter moving only in the vertical direction is given by p (t) = 4t ^3 − 23t ^2 + 15t. At what times is the helicopter motionless?

t = .36 and t = 3.5

With what initial velocity must an object be thrown upwards so that it will reach a maximum height of 800 feet? Assume the that the object experiences an acceleration of a (t) = −32 ft / sec2 due to gravity. Assume no air friction or other forces working on the object.

226 feet / sec

A student drops a penny off the top of a building. If the penny hits the ground 6 seconds later, how tall is the building?

y0 = 576 feet

John throws a baseball upwards at t = 0 with an initial velocity of 50 feet / sec. One second later, he throws another baseball upwards with an initial velocity of 20 feet / sec. Do the two balls ever cross each other before they hit the ground? Assume that the effect of gravity gives a constant acceleration of −32 feet / sec2 to each ball.

No, the balls do not cross each other

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AP Calculus AB: 10.1.3 Solving Vertical Motion Problems
TermDefinition

Solving Vertical Motion Problems

  • Given the initial velocity and initial position of an object moving vertically, you can use the fact that the acceleration due to gravity is a constant to find the velocity and position functions.

  • The velocity and position functions can be used to answer many questions about the motion of an object.

note

  • This problem asks you to use the length of time an object was falling to determine the distance it fell.

  • A falling object has no initial velocity since it is simply
    released, not thrown.

  • This fact simplifies the position function.
    You also know that the height when t = 5 is 0.

  • Evaluate the function at t = 5 to set up an equation you can solve.

  • Solve for the initial velocity y 0 to determine how far the object fell, which is also the depth of the pit.

  • In this situation you are asked to find the initial velocity
    needed to attain a given maximum height.

  • First you must find an expression for the time at which the object attains its maximum height. This will happen when the velocity is zero, so set the velocity function equal to 0 and solve for t in terms of v 0 .

  • Then substitute the expression for t into the position function and set the result equal to the maximum height of 25 and solve for v 0 . You will need to use your techniques for simplifying algebraic expressions to arrive at the answer.

  • An initial velocity of 40 ft/sec will give the object a maximum height of 25 ft.

The position of a helicopter moving only in the vertical direction is given by p (t) = 4t ^3 − 23t ^2 + 15t. At what times is the helicopter motionless?

t = .36 and t = 3.5

With what initial velocity must an object be thrown upwards so that it will reach a maximum height of 800 feet? Assume the that the object experiences an acceleration of a (t) = −32 ft / sec2 due to gravity. Assume no air friction or other forces working on the object.

226 feet / sec

A student drops a penny off the top of a building. If the penny hits the ground 6 seconds later, how tall is the building?

y0 = 576 feet

John throws a baseball upwards at t = 0 with an initial velocity of 50 feet / sec. One second later, he throws another baseball upwards with an initial velocity of 20 feet / sec. Do the two balls ever cross each other before they hit the ground? Assume that the effect of gravity gives a constant acceleration of −32 feet / sec2 to each ball.

No, the balls do not cross each other

The Martian lander “Land-o-matic” is designed to withstand a maximum impact velocity of 45 feet / sec. On one particular mission to Mars, the Land-o-matic is descending towards the Martian surface at a constant rate of −30 feet / sec when its engines abruptly fail. If the ship is 80 feet from the surface of the Mars at this instant, will it survive the impact when it hits the ground? Assume that Martian gravity gives the ship an acceleration of a (t) = −10 feet / sec2 .

No, the Land-o-matic doesn’t survive.

The mad scientist Dr. Experimento has decided to change the strength of gravity so that it will take 1 minute for a stone dropped from rest to fall 10 feet. Assume that the force of gravity gives the stone a constant acceleration a (t) = −G. Find the value of G in feet / sec2.

0.0056 feet / sec^2