# A projectile is shot from the ground at an angle of #(5 pi)/12 # and a speed of #3/5 m/s#. Factoring in both horizontal and vertical movement, what will the projectile's distance from the starting point be when it reaches its maximum height?

Given a particle's initial velocity, we are asked to determine how far it is from the launch point at the moment of its maximum height.

We can use the equations to accomplish this.

and

to determine the particle's maximum height in both vertical and horizontal directions.

The duration required for this to occur is provided by

Consequently, the separation from the launch site is

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To find the distance from the starting point when the projectile reaches its maximum height, you can use the formula:

[ \text{distance} = \text{initial velocity} \times \text{time at maximum height} \times \cos(\text{launch angle}) ]

First, find the time it takes for the projectile to reach its maximum height:

[ \text{time to maximum height} = \frac{\text{initial vertical velocity}}{\text{vertical acceleration}} ]

Then, use this time to calculate the horizontal distance:

[ \text{distance} = \text{initial horizontal velocity} \times \text{time to maximum height} \times \cos(\text{launch angle}) ]

[ \text{distance} = \left( \frac{3}{5} \right) \times \left( \frac{\frac{3}{5} \sin\left(\frac{5\pi}{12}\right)}{-9.8} \right) \times \cos\left(\frac{5\pi}{12}\right) ]

[ \text{distance} ≈ 0.166 , \text{m} ]

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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