Question Video: Recognizing the Relation between the Height and the Speed of a Vertically Projected Body | Nagwa Question Video: Recognizing the Relation between the Height and the Speed of a Vertically Projected Body | Nagwa

# Question Video: Recognizing the Relation between the Height and the Speed of a Vertically Projected Body Mathematics • Second Year of Secondary School

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If a body is projected vertically upwards with speed π to reach maximum height β, then the speed the body should be projected by to reach height 4β is οΌΏ. [A] π [B] 4π [C] 2π [D] β2π

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### Video Transcript

If a body is projected vertically upwards with speed π to reach maximum height β, then the speed the body should be projected by to reach height four β is blank. (A) π, (B) four π, (C) two π, (D) square root of two π.

Alright, so here we have a scenario where a body is projected vertically upward with a speed weβve called π. And under this influence, it achieves a maximum height β. We then imagine a scenario where this same body is projected to a height of four β. And the question is, with what initial speed would it need to be projected to reach this height? We have these four answer options here. And as we get started with our answer, we can notice the fact that this body, as it moves upward, is under the influence of only the force of gravity. Therefore, its acceleration is uniform, and we can describe its motion using an equation of motion. The equation weβll use is that an objectβs final velocity squared is equal to its initial velocity squared plus two times its acceleration times its displacement.

In the case of our vertically projected bodies, we can say that the final moment in time is the one where each one is at its maximum height. And in each case at this point, its velocity is zero. This means that the left-hand side of this expression will be zero. And then, as we fill in the right-hand side, letβs focus on this case where we have a maximum height β and an initial speed π. We would write then that zero is equal to π squared plus two times π, the acceleration our body undergoes, multiplied by β. Itβs possible to rearrange this equation so that it reads β is equal to negative π squared over two times π.

And in this instance, π is equal to negative 9.8 meters per second squared. And writing that value in without units, we see that the negative signs in numerator and denominator cancel. So if we want a body to ascend to a maximum height of β, we need to give it an initial speed of π. And if we want a body to ascend to a maximum height of four β, where we multiply both sides of the equation by four, then we can equivalently say that this is equal to two times π quantity squared divided by two times 9.8. And so, now we know what our objectβs initial velocity must be in order for it to be projected to a height of four times β. It must be twice the initial speed π.

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