# A projectile is shot from the ground at an angle of #pi/12 # and a speed of #8 /3 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?

At its maximum height, the projectile will have traveled a horizontal distance of

We want to calculate the range of the projectile at its maximum altitude.

Begin by breaking the initial velocity into components. Given that

Where

#pi/12=15^o#

Using basic trigonometry, we can see that the perpendicular (vertical,

#sin(theta)=(opp.)/(hyp.)#

#=>sin(theta)=(v_y)/(v)#

#cos(theta)=(adj.)/(hyp.)#

#=>cos(theta)=(v_x)/v#

We can rearrange to solve for

#v_y=vsin(theta)#

#=>v_y=(8/3m/s)sin(pi/12)#

#=>v_y~~0.69m/s# -

#v_x=vcos(theta)#

#=>v_x=(8/3m/s)cos(pi/12)#

#=>v_x~~2.58m/s#

At the projectile's maximum altitude, it will have

We can then put this value for

Calculating

#v_f=v_i+a_yDeltat#

#Deltat=(v_f-v_i)/a_y#

#Deltat=(0-0.69m/s)/(-9.8m/s^2)=0.070s# Now we calculate how far the projectile has traveled horizontally (i.e. the range) in this amount of time. Remember that there is no horizontal acceleration.

#Deltax=v_iDeltat+cancel(1/2aDeltat^2)#

#Deltax=(2.58m/s)(0.070s)~~0.18m#

#:.# At its maximum height, the projectile will have traveled a horizontal distance of#0.18m# from the launch point.

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The horizontal distance ((D)) traveled by the projectile at its maximum height can be calculated using the formula:

[ D = \frac{{v_0^2 \cdot \sin(2\theta)}}{g} ]

Given:

- Launch angle ((\theta)): (\frac{\pi}{12})
- Initial speed ((v_0)): (\frac{8}{3}) m/s
- Gravitational acceleration ((g)): (9.8 , \text{m/s}^2)

Substitute the values into the formula:

[ D = \frac{\left(\frac{8}{3}\right)^2 \cdot \sin\left(2 \times \frac{\pi}{12}\right)}{9.8} ]

Calculate to find the horizontal distance ((D)).

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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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