# A projectile is shot from the ground at an angle of #pi/4 # and a speed of #10 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?

The distance is

Applying the equation of motion

Therefore,

Applying the equation of motion

The distance travelled in the horizontal direction is

The distance from the starting point 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 \text{Cosine of launch angle} ]

Given:

- Initial velocity (( v_0 )) = 10 m/s
- Launch angle (( \theta )) = ( \frac{\pi}{4} )
- Time to reach maximum height (( t_{\text{max}} )) = ( \frac{v_0 \times \sin(\theta)}{g} ) (where ( g ) is the acceleration due to gravity, approximately ( 9.8 , \text{m/s}^2 ))

Substituting the given values into the formula:

[ t_{\text{max}} = \frac{10 \times \sin\left(\frac{\pi}{4}\right)}{9.8} ] [ t_{\text{max}} = \frac{10 \times \frac{\sqrt{2}}{2}}{9.8} ] [ t_{\text{max}} = \frac{5\sqrt{2}}{9.8} ]

Now, substitute ( t_{\text{max}} ) into the distance formula:

[ \text{Distance} = 10 \times \frac{5\sqrt{2}}{9.8} \times \cos\left(\frac{\pi}{4}\right) ]

[ \text{Distance} \approx 5 , \text{meters} ]

So, the distance from the starting point when the projectile reaches its maximum height is approximately ( 5 , \text{meters} ).

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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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