# How is speed affected (or not affected) by acceleration by gravity?

I am a new physics student and I was reading a question posted on here earlier:

Which Sphere is Fastest?

It posits that three spheres thrown with the same speed at the same point (one thrown horizontally, one thrown vertically up, and one thrown vertically down) all hit the ground at the same speed.

Doesn't the one which is in the air the longest gain the most speed? beacuse of this : 9.8m/s2

You can see that I am very new to physics and this website. I'll be asking dumb questions and all help is appreciated. Any tips about other resources and help is welcome.

• When you get to conservation of energy in your physics course, ask your instructor the same question. – David White May 3 at 17:30

If we look at he scalar value of (9.8m/s)/s, then it does appear that a ball that is thrown upwards will be in the air longer and therefore will have a larger s and therefore a larger speed. However, when we look at acceleration as a vector, we see that for the ball thrown upwards, the velocity starts out in the direction opposite that of the acceleration. So rather than being the air causing the speed to increase, for the first part of its flight gravity will be decreasing its speed. It will then fall back down, and the time that it takes to fall back down to its original height will be equal to the time it took going upwards. So the amount that its speed decreased going up will be equal to the amount that the speed increases going down. So when the ball thrown upwards reaches its original height, it will have the same velocity as the ball thrown downwards originally had.

• Right! So the speed I throw the ball with is the same speed the ball possesses when it gets back to the point where the other two balls were thrown initially. Thank you. – Justinfromthewest May 3 at 18:35

The motion of the spheres can be derived from motions in two perpendicular and independent paths viz., horizontal and vertical axes.

The motion along the horizontal axis is independent of the motion along the vertical axis and vice-versa.

But the resultant path is different, it may be parabolic, simply horizontal or even vertical.

For example, let's say we roll a ball P at a speed of 10 m/s (assuming zero friction) along the horizontal axis and through another ball Q with a speed of 10 m/s along the horizontal axis and 25 m/s along the vertical axis. Let us say ther is no acceleration in horizontal direction ( but g acts along vertical direction).

If we find the ball P to have traversed 10 m after a second, we will also find the Q has also traversed 10 m horizontally, however there is a vertical displacement between the two.

The sphere that is thrown up will first loose its speed, than regain it as it falls down. So even if it is longer in the air, the resultant speed doesnt need to be bigger (and it will be the same as you mentioned).

Similarly the horizontally thrown ball - it is longer in the air than the one that is thrown vertically down, but because of pythogorean theorem it needs to gain bigger vertical velocity to produce same speed as the ball that was thrown down and in the result, the speed will be same.

It is not as simple as you think. Acceleration is defined as rate of change of velocity. Now , acceleration is a vector. So it is not so simple. You have to consider if velocity and acceleration are in same direction or not (for 1-d motion) or even deal with vectors(for motion in 2-D). It doesn't make any sense by simply saying: which spends more time in air! Imagine an object thrown vertically upward from ground and another object dropped from height. Even for same time, the speed of one decreases while for other, increases . In one case, it is thrown horizontally, so acceleration and initial velocity are perpendicular, so you have to consider motion in 2-D. It is the complete set of equations of kinematics by which you can judge the answer. But on the other hand, there is something like energy conservation; which can be successfully applied here to get desired result. Which says that if initial energies are same, so will their final energies (which is only the form of kinetic energy). By this argument, no rigorous calculation is required and simplifies approach. Hope this helps you if you have read it to the end!

Doesn't the one which is in the air the longest gain the most speed? beacuse of this : 9.8m/s2

If you throw a sphere up and it comes back to where it started from the sphere gains no kinetic energy so its speed must be the same as when it started.
So that extra time spent by the sphere going up and then coming down to where it started from gains it no kinetic energy.

The sphere which is thrown downwards certainly takes least time to reach the ground but it still only gains the same amount of energy $$mgH$$ (change in gravitational potential energy) as the other two spheres.

The important thing is that all three spheres started with the same amount of kinetic energy $$\frac 12 mv^2$$ and lost the same amount of gravitational kinetic energy $$mgH$$ so they had the same amount of kinetic energy at the end of travels.

Compare this will a mass sliding down a frictionless slope and another mass falling vertically down the same height.
The vertical fall will take a shorter period of time but the masses will arrive at the bottom of the slope with the same kinetic energy having lost the same amount of gravitational potential energy.

• @annav Thank you. You look after me in an exemplary way! – Farcher May 3 at 16:55

As you study Physics you will learn how to discern one thing from another, and look at all the different aspects of the problem. The force of throwing the sphere downward was independent of gravity and therefore time in the air. You have to add the two together. Then, the sphere thrown upward, with the same force as the one downward, is going to fall from a higher point after it comes to a stop which will be equal to the one thrown downward because when it gets back to the point of being thrown it's going to be dropping at the same speed as the one that was thrown downward. The one thrown horizontally will not have the benefit of that extra downward speed.