GRED: Does a ball take longer to go up or come down?

[RJN]

Guess the Result of the Experiment of the Day (GRED): A Ball Tossed Vertically

Category: Physics; Classical Mechanics

The answer is: Down. Only if there was no air resistance would the up and down times be the same. Although the above poll (now closed) did not indicate the correct answer, many of the below posters explained it well. Please scroll down to read why it takes longer for a ball to come down than go up.

- RJN

[neufer]

• • __ _The Brave/Valiant Little Tailor_ by The Brothers Grimm
    ____ translated by Margaret Taylor (1884)
  <<The giant picked up a stone and threw it so high that the eye could scarcely follow it. "Now, little mite of a man, do that likewise." "Well thrown," said the [brave/valiant little] tailor, "but after all the stone came down to earth again; I will throw you one which shall never come back at all." And he put his hand into his pocket, took out the bird [that he had rescued], and threw it into the air. The bird, delighted with its liberty, rose, flew away and did not come back. "How does that shot please you, comrade?" asked the tailor.>>

[Chris Peterson]

This is a classic calculus problem- one that isn't real easy to solve analytically. It can be examined intuitively, and in this case intuition and formal calculation arrive at the same result: the ball takes longer to fall than it does to rise. Viewed qualitatively, drag is acting downwards while the ball is rising, which reduces the time the ball spends traveling upwards; drag is acting upwards while the ball is falling, which increases the time the ball spends traveling downwards.

Another way to view this intuitively is to consider a more extreme case: shooting a bullet upwards. The bullet can be launched very fast- a supersonic speed, even. But from the zero-velocity peak of its flight, it can not fall any faster than terminal velocity. So it will clearly take longer coming down than it did going up.

The question might be framed a little more rigorously, since you probably want to exclude the part of the throw where the ball is being accelerated, and consider the time from the end of the throw to the peak, and the time from the peak back to the release height. Probably not a bad idea to make it clear you are talking about doing this experiment in an atmosphere. The answer is different if you do it in a vacuum.

[BMAONE23]

But a Bullet's muzzle speed is much faster than its tumbling terminal velocity so a Bullet should take longer to fall than it does to reach maximum altitude. The Ball would have several dependancies with the most important being the height attained with the throw. The higher it travels, the more likely it will reach terminal velocity on its return trip. Since no human can throw a ball at 200+ mph it would likely return faster if the thrown altitude allows for a faster return speed.

[neufer]

Probably not a bad idea to make it clear you are talking about doing this experiment in an atmosphere.
The answer is different if you do it in a vacuum.

Of course, they did make it clear that they were talking about doing the experiment in an atmosphere:

"A ball is tossed in the air."

They should have made it clear that gravity was involved and that the ball wasn't being bounced off the floor:

A ball is tossed UP in the air.

[neufer]

Since no human can throw a ball at 200+ mph it would likely return faster if the thrown altitude allows for a faster return speed.

Or just throw a Lawn dart:

<<Lawn darts (also called Jarts) is a lawn game for two players or teams. A lawn dart set usually includes four large darts and two targets. The game play and objective are similar to both horseshoes and darts. The darts are similar to [but probably more deadly than] the ancient Roman plumbata. They are typically 12 inches (30 cm) long with a weighted metal or plastic tip on one end and three plastic fins on a rod at the other end. The darts are intended to be tossed underhand toward a horizontal ground target, where the weighted end hits first and sticks into the ground. The target is typically a plastic ring, and landing anywhere within the ring scores a point.>>

<<Plumbatae or martiobarbuli were lead-weighted darts carried by infantrymen in Antiquity and the Middle Ages. The first examples seem to have been carried by the Ancient Greeks from about 500 B.C. onwards, but the best-known users were the late Roman and Byzantine armies.>>

[neufer]

This is a classic calculus problem- one that isn't real easy to solve analytically. It can be examined intuitively, and in this case intuition and formal calculation arrive at the same result: the ball takes longer to fall than it does to rise. Viewed qualitatively, drag is acting downwards while the ball is rising, which reduces the time the ball spends traveling upwards; drag is acting upwards while the ball is falling, which increases the time the ball spends traveling downwards.

In other words, from the top of flight:

1) Time played forward : gravity & drag forces are in opposite directions so it takes a longer time to reach the ground.
2) Time played backward : gravity & drag forces are in the same direction so it takes a shorter time to reach the ground.

One can find various ways of modifying drag in either case (tumbling bullets, jarts, etc.) but it can never be made negative.

[waywardson]

i thought of it as an energy problem. as the ball travels on its arc, air resistance is constantly being reduced, which means that it's maximum kinetic energy is reduced both the start and end of the trip down being lower than the trip up. I initially said that it would take longer going down due to this reduction in energy, but this is an integration over the arc so the reduction could balance out on both sides. I'm working on the math but it's wierd, I'm thinkin difeq right now, might be able to get around it though.
And the problem says the ball is thrown in the air, and since its throw terminal velocity won't come into play.

[RogerSB]

Drag should be the same going up or down. Otherwise, parachutists would be in for a quick trip. If it actually isn't, what causes the difference?

[Chris Peterson]

Drag should be the same going up or down. Otherwise, parachutists would be in for a quick trip. If it actually isn't, what causes the difference?

The magnitude of the drag force is dependent only on speed, and is independent of the direction the ball is moving. But the direction of the drag force isn't the same for both cases. So you have an inherent asymmetry in the relevant forces: gravity is always downward directed, but drag is downward directed when the ball is moving up, and upward directed when the ball is moving down.

[Chris Peterson]

i thought of it as an energy problem. as the ball travels on its arc, air resistance is constantly being reduced, which means that it's maximum kinetic energy is reduced both the start and end of the trip down being lower than the trip up...

The problem becomes more complex if you add a horizontal velocity component. I assumed that the ball is thrown straight up, so everything happens along a single axis (i.e. no "arc", unless it is an arc of position versus time).

[Beta]

The higher [the ball] travels, the more likely it will reach terminal velocity on its return trip. Since no human can throw a ball at 200+ mph it would likely return faster if the thrown altitude allows for a faster return speed.

You are imposing a limit on how fast one can throw a ball, but no limit on how high one can throw it.

The ball cannot return with more energy than it had when it started. If you can't throw it that fast, you can't throw it high enough to come back that fast.

[Feagles]

Clearly, the answer depends on how the ball was thrown. If it was catapulted faster than it's terminal velocity on return, it will take longer to come down. If less than terminal velocity it should be the same. I was tempted to add that where it was being thrown could enter in as well. If it's on the space shuttle .... But since up and down are so subjective/irrelevant I decided to let that go. On the other hand has anyone checked to see if one could hand toss a ball on a lower gravity celestial body with an atmosphere fast enough to reach terminal velocity there? But that atmosphere would probably not be called air.

[Chris Peterson]

Clearly, the answer depends on how the ball was thrown.

I don't think it matters at all how the ball is thrown. Regardless of how fast or how slow the ball is launched, it will always take longer to come down than it did to go up.

[BMAONE23]

Perhaps it is time to get the mythbusters involved Or Nolan Ryan and a Radar speed detector.

[Hoebeau]

Without any further information, you must consider the scenario where the ball is thrown upwards faster than terminal velocity. When it stops rising under gravity's downward pull, it will then accelerate downward until it reaches terminal velocity, which is slower than its starting upward velocity in this scenario. Thus, it will take longer to fall the same distance it rose.

[biddie67]

I think that it will take longer to come down. At each point in altitude, the combined forces of gravity and drag will be the same on both the trip up and the trip down (unless the barametric pressures change inbetween the two trips). It seems that the big factor is the force used to toss the ball up to start with. It will go up at the velocity (and length of time embedded in it) as it overcomes gravity and drag and is exhausted of that residual of energy. But on the way back down, it would seem to be only at the mercy of the forces of gravitational acceleration which might be less than the original force to toss it so the time to come back down would be longer ....

[MurryClan]

It is not said how the ball is thrown. If it is thrown by me, The ball would not have enough speed to reach an altitude to obtain terminal velocity. So I figure it would take the same time going up as down. As I said it was not stated how the ball was thrown and one could make a way to have that ball thrown at speeds greater than terminal velocity, in which case it would take the ball longer to come down.

[WrB]

Think of the same situation in a vacuum. The balls initial velocity determines the balls maximum height (from the equation of movement in a vacuum). I would presume that it would be equal time in this case.

But in atmosphere, the ball has drag and the ball never reaches this calculated height (from our equation of movement in a vacuum). Some of that kinetic energy is lost due to drag. At the top of the arc, the balls potential energy is not equal to the balls initial kinetic energy because of these losses. On the way down, as the ball crashes into the ground, the balls downward speed will be less than the balls initial upward speed, due to the drag exerted on the ball during both the upward and downward legs of this trip. Thus the average speed upward is greater than the average speed downward. So I would say that it takes longer for the ball to fall.

[WrB]

Think of the same situation in a vacuum. The balls initial velocity determines the balls maximum height (from the equation of movement in a vacuum). I would presume that it would be equal time in this case.

But in atmosphere, the ball has drag and the ball never reaches this calculated height (from our equation of movement in a vacuum). Some of that kinetic energy is lost due to drag. At the top of the arc, the balls potential energy is not equal to the balls initial kinetic energy because of these losses. On the way down, as the ball crashes into the ground, the balls downward speed will be less than the balls initial upward speed, due to the drag exerted on the ball during both the upward and downward legs of this trip. Thus the average speed upward is greater than the average speed downward. So I would say that it takes longer for the ball to fall.

Correction: "as the ball crashes into the ground" needs to be "as the ball approaches it's initial height", due to the fact that we are not throwing the object from ground level, but slightly above it.

[Amir]

at first i voted for "Same", cuz i assumed that the ball is thrown in vacuum. but when you all said that it's in atmosphere, i took another look: "...in the air..."! so i'll say "down".
i agree with Chris & biddie67:

...Viewed qualitatively, drag is acting downwards while the ball is rising, which reduces the time the ball spends traveling upwards; drag is acting upwards while the ball is falling, which increases the time the ball spends traveling downwards. ...

we throw the ball so we give it a Kinetic energy (K₁), which drag force (fk₁) will reduce it while it changes to Potential energy (U): U = K₁- fk₁
when the ball is coming back drag force will do the same thing again while Potential energy changes to Kinetic energy (K₂): K₂ = U - fk₂
thus K₂ < K₁, same mass --> V₂ < V₁, same distance --> t₂> t₁

[neufer]

[list]ƒ = integration

T_u = total time up
T_d = total time down

V_u(z) = Velocity up at a given height z
V_d(z) = Velocity back down at a given height z
V_u(z) ≥ V_d(z) due to any air resistance in the interim

T = ƒ dt = ƒ dz/(dz/dt) = ƒ dz/V(z)

T_u = ƒ dz/V_u(z) : T_d = ƒ dz/V_d(z)

V_u(z) ≥ V_d(z) due to any air resistance in the interim

Thus: 1/V_u(z) ≤ 1/V_d(z)

Ergo: T_u ≤ T_d : { T_u = T_d if and only if air resistance = 0 }[/list]

[Bert and Ernie]

If going up reaches escape velocity, it takes longer to go up.

[neufer]

If going up reaches escape velocity, it takes longer to go up.

If going up reaches escape velocity, please call the Washington Nationals immediately!

[dog]

When the ball moves in the atmosphere, some of the air is pushed along with it (drag). As the ball continually slows down on its upward flight, some of this moving air will push on the ball. When the ball reverses direction and falls, it must now move through this rising stream of air. The ball must take longer falling as it must also overcome the greater air distance and resistance.