Why does an object fall in the same time as it takes to reach a peak

In summary: So the time taken for the downward trip will be the same as the time taken for the upward trip, because the distance is the same and the acceleration is the same. This is demonstrated by the equations provided, showing that the time taken is 1.5 seconds. In summary, when an object jumps a certain height and is under the influence of the same acceleration, the time it takes for the upward and downward trips will be the same due to the conservation of energy. This can be observed in equations of motion, but may not always hold true in real-world scenarios due to complicating factors.
  • #1
mikeza
2
0

Homework Statement



A kangaroo jumps a height of 2.8m. How long was it in the air?

Homework Equations



v=v0+at

x=x0+v0t+(1/2)at^2

v^2=v0^2+2aΔx

The Attempt at a Solution



-2.8 = -4.9t^2
.57 = t^2
t = .76 s (down)

0 = v0^2+2(9.8)(2.8)
54.88 = v0^2
v0 = 7.408

0 = 7.408-9.8t
t = .76 (up)

t = 1.5 sec

SO...
I know I have the answer right I just don't understand how that works conceptually - that if something has a initial velocity upwards it takes the same time as if the initial velocity was 0 downwards. How can something traveling the same distance (just one negative and one positive) with the same acceleration but different initial velocities take the same amount of time out to the 9th decimal??
 
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  • #2
The thing initially has some kinetic energy. As it moves up, that kinetic energy is converted into gravitational potential energy, and then as it comes back down it is converted back into kinetic energy. Since energy is conserved, it would make sense that the initial velocity before leaving the ground is the same as just before it touches back down.

It doesn't work that way in the real world though, what with air resistance doing negative work on the way up and the way down, among other things.
 
  • #3
Thanks a bunch. Can't wait to teach my class tomorrow... my teacher doesn't tell us anything.
 
  • #4
Excellent observation, though you are not the first to make it. In fact it is much discussed you will later find.

The equations you quote are applications of Newton's equations of motion. These are reversible, i.e. if you exactly reverse the direction of every particle is traveling giving it the same velocity but in the exact opposite direction it will retrace the exact path it came by. People talk of substituting -t for t in equations and the solution is sort of the same in both cases but with time reversed.

To a extent this happens and can be observed in reality - as you have just done really. The reason it does not work in practice is also much discussed, and is that if the situation is more than so much complicated, you cannot realize that reversal. That is if you drop a drop of ink into water, according to the equations (which we do not have to solve, it is just a general property they have) if you could get behind each ink particle and for that matter all the water molecules, and push them back so as each have exactly equal opposite velocity they were traveling in that instant, then they would retrace their motions in reverse, reconcentrate together into an ink drop which would come out of the water back into the dropper. But you can see the 'if you could' is not a reasonable expectation, which explains why you will never see this happen.

If you are starting science expect to hear more of this.
 
Last edited:
  • #5
mikeza said:

Homework Statement



A kangaroo jumps a height of 2.8m. How long was it in the air?

Homework Equations



v=v0+at

x=x0+v0t+(1/2)at^2

v^2=v0^2+2aΔx

The Attempt at a Solution



-2.8 = -4.9t^2
.57 = t^2
t = .76 s (down)

0 = v0^2+2(9.8)(2.8)
54.88 = v0^2
v0 = 7.408

0 = 7.408-9.8t
t = .76 (up)

t = 1.5 sec

SO...
I know I have the answer right I just don't understand how that works conceptually - that if something has a initial velocity upwards it takes the same time as if the initial velocity was 0 downwards. How can something traveling the same distance (just one negative and one positive) with the same acceleration but different initial velocities take the same amount of time out to the 9th decimal??

It may have been deliberate, but the "give away" equation you did not list [or was not stated] is

x = (vo + vf)/2 x time.

On the way up, the object starts at some velocity and finishes with zero velocity, thus covering a certain distance on the way. The time was how ever long it takes to change your velocity from the original value to zero, while acceleration is g.
On the way down, it has to cover the same distance, under the influence of the same acceleration [g] so will start at velocity 0, and finally reach the same speed as it was projected at, but traveling down not up.
 

1. Why do objects fall at the same rate?

Objects fall at the same rate because of the force of gravity. Gravity is a force that pulls all objects towards the center of the Earth. As long as there is no air resistance, all objects will fall at the same rate regardless of their mass.

2. Is there any variation in the time it takes for an object to fall?

No, there is no variation in the time it takes for an object to fall. This is because the acceleration due to gravity is constant and independent of mass. This means that all objects will accelerate at the same rate and fall in the same amount of time.

3. How does air resistance affect the time it takes for an object to fall?

Air resistance can affect the time it takes for an object to fall. Objects with a larger surface area, like a feather, will experience more air resistance and take longer to fall than objects with a smaller surface area, like a rock. However, in a vacuum where there is no air resistance, all objects will still fall at the same rate.

4. What is the relationship between an object's mass and the time it takes to fall?

The mass of an object does not affect the time it takes to fall. As mentioned earlier, the acceleration due to gravity is constant and independent of mass. This means that all objects, regardless of their mass, will fall in the same amount of time.

5. Why do objects reach their peak height at the same time it takes to fall?

Objects reach their peak height at the same time it takes to fall because of the conservation of energy. When an object is thrown or dropped, it has potential energy due to its height. As the object falls, this potential energy is converted into kinetic energy. At the peak height, all potential energy has been converted into kinetic energy, resulting in the same time for the object to fall back to its starting point.

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