When and How Do They Land?

From the top of a tower, one ball is dropped and another thrown horizontally with a velocity v v . How will they reach the ground?

Assumptions and Details

  • Neglect the rotation of the Earth.
Simultaneously with same speed Simultaneously with different speeds At different times with same speed At different times with different speeds

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2 solutions

Both situations have an initial speed, u u , of 0.
But let's say that the height is 10 meters

You can use the "suvat" equations to solve for both.

For the vertical ball:
v 2 = u 2 + 2 a s v = = 0 2 + 2 a s = 2 9.81 10 14.00 m/s \begin{aligned} v^2 &=& u^2 + 2as \\ v &=& = \sqrt{0^2 + 2as} \\ &=& \sqrt{2 \cdot 9.81 \cdot 10} \\ &\approx& 14.00 \text{ m/s} \end{aligned}

And to find the time taken:

v = u + a t = 0 + a t t = v a = 14 / 9.81 1.43 s \begin{aligned} v &=& u + at \\ &=& 0 + at \\ t &=& \frac va \\ &=& 14/9.81 \\ &\approx& 1.43 \text{ s} \end{aligned}

For the horizontal ball,
there are 2 components therefore it is a different speed
v x 2 = u x 2 + 2 a s x v^2_{x}=u^2_{x} + 2as_{x} horizontal component.
v y 2 = u y 2 + 2 a s y v^2_{y}=u^2_{y} + 2as_{y} vertical component.

I think the question is worded poorly. As I recall in school the always used the term "just before striking the ground" This prevents the confusion that the object stops when it hits the ground. Both objects stop when they " reach the ground!"

Preston Wigfall - 5 years, 3 months ago

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But I wrote it as "will reach the ground". It has the same meaning as "about to strike", right?

Anandhu Raj - 5 years, 3 months ago
Anandhu Raj
Feb 2, 2016

In both cases, initial vertical component of velocity is zero. So both reaches the ground simultaneously but with different speeds as in second case there is horizontal component of velocity.

i see now, thanks

Mateo Ceballos Querol - 5 years, 3 months ago

Nothing was said in the problem as stated about ignoring the Earth's curvature. The ball with horizontal component will reach the surface later and at a greater speed. In order to meet the criteria of making an accurate scientific statement, be very careful to be precise in your wording. That said, there was also nothing said about the initial speed and that could be stated as being great enough to escape Earth's gravitational field, in which case that ball would not only not hit the ground, it would be on its merry way into the solar system. Where is unimportant. I want a credit for the "Oops" I was given, when in fact I was correct based upon the available information. Admittedly, I limited the horizontal component so that ball would also strike the surface.

J. Richard Jacobs - 4 years, 10 months ago

so you want to say if i drop bullet from 10m and in same time i shoot bullet horizontal two of thouse bullets will hit ground at same time

Calvin edited the comment.

Ilija Stevanovic - 5 years, 4 months ago

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Considering that there's no air resistance and ignoring the curvature of the Earth, yes, they will reach the ground at the same time.

If there's air resistance they experience different magnitude of air resistance depending on the medium they travel.

Kenneth Choo - 5 years, 4 months ago

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