Find Initial Velocity given angle, distance, height launched from.

In summary, to find the initial velocity of a nerf gun fired in a vacuum, we can use the known values of g, yo, xo, y, x, and θ and the equations x=xo+vot, y=yo+vot+.5gt2, vox=vocosθ, and voy=vosinθ. By simplifying the x equation and substituting in the y equation, we can find the initial velocity and its horizontal and vertical components.
  • #1
mhsphysics10
2
0
1. A nerf gun in a vacuum is fired from .98 meters at an angle of 15degrees, going 7.86m. Find the initial velocity of the gun, then find the horizontal and vertical components of the initial velocity.

Knowns
g=-9.8m/s2
yo
=.98m
xo=0m
y=o ( the final height of the dart is 0)
x=7.86m
θ=11 degrees



Homework Equations


x=xo+vot
y=yo+vot+.5gt2
vox=vocosθ
voy=vosinθ

The Attempt at a Solution


So first i simplified the x equation down to get time on one side to get
(x-xo)/(vocos11)=t
Then i plugged in the numbers to get
{7.86/(vocos11)}=t

Then i plugged in the parts for the y equation,
-.98=voyt+-4.9t2
Now what do i do? do take out a t and plug in the x equation?
 
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  • #2
Why not substitute in v_0* sin (theta) = v_0y. Then you have two equations and two unknowns and from that everything else is easy to find!
 
  • #3


To find the initial velocity, we can use the fact that the time taken for the dart to reach its maximum height and then return to the ground is the same for both the horizontal and vertical components. This is because there is no acceleration in the horizontal direction.

So, we can set the two equations equal to each other and solve for the initial velocity (vo):

(x-xo)/(vocos11) = -.98/(vosin11) + .5(-9.8)t

Simplifying and plugging in the values, we get:

(7.86 - 0)/(vo*cos11) = -.98/(vo*sin11) + .5(-9.8)t

7.86/(vo*cos11) = -.98/(vo*sin11) - 4.9t

Multiplying both sides by vo, we get:

7.86*sin11 = -0.98*cos11 - 4.9t*vo

Solving for vo, we get:

vo = 7.86*sin11/(-0.98*cos11 - 4.9t)

Plugging in the values, we get the initial velocity to be approximately 5.87 m/s.

To find the horizontal and vertical components, we can use the equations:

vox = vo*cos11 = 5.87*cos11 = 5.78 m/s

voy = vo*sin11 = 5.87*sin11 = 1.33 m/s

So, the horizontal component of the initial velocity is 5.78 m/s and the vertical component is 1.33 m/s.
 

What is the formula for finding initial velocity given angle, distance, and height launched from?

The formula for finding initial velocity is v0 = √(g*d tanθ / 2h), where v0 is the initial velocity, g is the acceleration due to gravity (9.8 m/s²), d is the horizontal distance travelled, θ is the launch angle, and h is the initial height.

How do I determine the launch angle if I know the initial velocity, distance, and height launched from?

The launch angle can be determined using the formula θ = 0.5 * arctan(2hv0²/gd), where θ is the launch angle, h is the initial height, v0 is the initial velocity, and d is the horizontal distance travelled.

Can I use this formula for any type of projectile motion?

No, this formula is specifically for finding the initial velocity of a projectile launched at an angle from a given height. Other types of projectile motion, such as horizontal or vertical launches, may require different formulas.

What units should I use for the variables in the formula?

For best results, it is recommended to use consistent units throughout the formula. The initial velocity (v0) should be in meters per second (m/s), the distance (d) in meters (m), the height (h) in meters (m), and the launch angle (θ) in degrees (°).

Can I use this formula to find the initial velocity for any distance or height?

Yes, this formula can be used to find the initial velocity for any distance and height launched from, as long as the launch angle is known. However, for shorter distances and lower heights, the initial velocity may be too small to accurately measure.

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