Vertical Ball Movement: Calculating Time in Flight (Calculus Practice)

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In summary, the conversation revolves around a calculus question involving the integration of a ball's velocity and acceleration. The given information includes the acceleration due to gravity and the initial velocity of the ball. The person is seeking tips for solving this type of problem, but it is mentioned that studying the night before a test may not be the most effective method.
  • #1
VikingStorm
[SOLVED] Integration (story)

Use a(t) = -32 feet per second squared as the acceleration due to gravity. A ball is thrown vertically upward from the ground with an initial velocity of 56 feet per second. For how many seconds will the ball be going upward?

(Calculus question)

I don't see where I would pull in calculus into this. (Will be on Calc test tomorrow, so other tips to dissect these kind of problems would be nice)
 
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  • #2
Well, what do you know that you can apply to this?

(P.S. the night before is generally a little late to be studying for a test; you'd probably be better off getting a good night's rest)
 
  • #3


Hi there,

In order to solve this problem using calculus, we can use the equation d(t) = -16t^2 + v0t + d0, where d(t) is the height of the ball at time t, v0 is the initial velocity, and d0 is the initial height (in this case, 0). We can use this equation to find the time at which the ball reaches its maximum height, which would be the time at which it stops going upward and starts falling back down.

To find this time, we can take the derivative of the equation with respect to time, which gives us v(t) = -32t + v0. Setting this equal to 0 and solving for t, we get t = v0/32. Plugging in the given initial velocity of 56 feet per second, we get t = 56/32 = 1.75 seconds.

Therefore, the ball will be going upward for 1.75 seconds before it starts falling back down. I hope this helps and good luck on your calculus test tomorrow! Remember to always break down the problem into smaller parts and use the appropriate equations and techniques.
 

FAQ: Vertical Ball Movement: Calculating Time in Flight (Calculus Practice)

What is vertical ball movement?

Vertical ball movement refers to the motion of a ball moving in a straight line up or down, perpendicular to the ground.

What is the formula for calculating time in flight?

The formula for calculating time in flight is t = √(2h/g), where t is the time in seconds, h is the height in meters, and g is the acceleration due to gravity (9.8 m/s^2).

How is calculus used in calculating time in flight?

Calculus is used to find the derivative of the position function of the ball, which represents its instantaneous velocity. By setting this derivative equal to zero, we can find the maximum height of the ball, which is then used in the formula for time in flight.

What are some real-world applications of calculating time in flight?

Calculating time in flight is useful in a variety of sports, such as basketball, baseball, and tennis, as it helps determine the trajectory of a ball and predict the time it will take to reach its target. It is also important in physics and engineering, where understanding the motion of objects is crucial.

How does air resistance affect vertical ball movement?

Air resistance can affect vertical ball movement by slowing down the ball's velocity and altering its trajectory. This can lead to a shorter time in flight and a lower maximum height. The amount of air resistance depends on the shape and size of the ball, as well as the air density and velocity.

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