Find Projectile Flight Time Given Only Maximum Height

In summary, the help section suggests that you don't need the horizontal displacement, and can figure out the flight time of a projectile with just its maximum height, but there are lots of combinations of angles and velocity that will produce the same maximum height with different flight times.
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Homework Statement
A projectile reaches a maximum height of 14.5 meters and travels a horizontal distance of 52.6 meters. How long was it in the air?
Relevant Equations
Vfy = V0y - 9.8t
I've been trying some online projectile problems. Specifically, I was using this one on master difficulty, looking at row e. It uses a random number generator; I shared the data I received in the Homework Statement above.

According to the help section, you can solve this with the formula given, but I don't see how. In fact, they seem to suggest that you don't even need the horizontal displacement, and can figure out the flight time of a projectile with just its maximum height, but there are lots of combinations of angles and velocity that will produce the same maximum height with different flight times.

I sort of cheated my way into a solution using this calculator. Note the calculator can't solve for time with just the two pieces of data given. What I did was put in horizontal distance (and initial height of 0) and start guessing at times until I found that 3.44 seconds produces a height of 14.5 meters.

Can anyone explain to me the correct approach to solving this problem?
 
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  • #2
trryan5 said:
In fact, they seem to suggest that you don't even need the horizontal displacement, and can figure out the flight time of a projectile with just its maximum height,
That is correct.
trryan5 said:
but there are lots of combinations of angles and velocity that will produce the same maximum height with different flight times.
Why do you think this?

Hint: Analyze just the vertical component of the motion. You can play with the kinematic formulas until you can solve it.

Hint2: For an object dropped from that height, how long would it take to fall?
 
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  • #3
trryan5 said:
... figure out the flight time of a projectile with just its maximum height, but there are lots of combinations of angles and velocity that will produce the same maximum height with different flight times.
Consider these three baseball throws:
1654194693973.png


Knowing nothing else, at a guess, do you expect the three of them to have different flight times?

What if I told you they were not three baseball throws, but only one, just viewed from different locations on the field? What would you say about flight time now?
 

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Doc Al said:
That is correct.

Why do you think this?

Hint: Analyze just the vertical component of the motion. You can play with the kinematic formulas until you can solve it.

Hint2: For an object dropped from that height, how long would it take to fall?
Ah, now I feel foolish. I was so focused on not knowing the initial upward velocity that I didn't think about analyzing it as an object in free fall.
 
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  • #5
trryan5 said:
Ah, now I feel foolish. I was so focused on not knowing the initial upward velocity that I didn't think about analyzing it as an object in free fall.
You could also consider two frames of reference. One at rest relative to the ground. And the other moving with the horizontal velocity of the projectile. The time of flight is the same in both (as it must be), yet the horizontal velocity and displacemet are different in each.
 
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1. What is the formula for finding the flight time of a projectile given only its maximum height?

The formula for finding the flight time of a projectile is t = √(2h/g), where t is the flight time, h is the maximum height, and g is the acceleration due to gravity (9.8 m/s²).

2. How do I determine the maximum height of a projectile?

To determine the maximum height of a projectile, you can use the formula h = (v₀²sin²θ)/2g, where h is the maximum height, v₀ is the initial velocity, θ is the angle of projection, and g is the acceleration due to gravity.

3. Can I use this formula for any type of projectile?

Yes, this formula can be used for any type of projectile as long as it is launched at an angle and the maximum height is known.

4. What units should I use when plugging values into the formula?

The units should be consistent throughout the formula. For example, if the maximum height is given in meters, then the acceleration due to gravity should also be in meters per second squared (m/s²).

5. Is there a way to calculate the flight time without knowing the maximum height?

Yes, there is another formula that can be used to calculate the flight time without knowing the maximum height. It is t = (2v₀sinθ)/g, where t is the flight time, v₀ is the initial velocity, θ is the angle of projection, and g is the acceleration due to gravity.

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