What is the initial velocity of a projectile on a sloped surface?

In summary, a projectile needs to hit a target on a hill slope with a line described by the equation Y=.4bX. The projectile is launched at an angle of 60° with an initial distance of 60m between the two points. Using the equations x = v_o cos60 t, y = v_o sin60 t - \frac{1}{2} a t^2, and a^2 + b^2 = c^2, the initial velocity of the projectile can be found in terms of b. After solving for x and y in terms of b, the final equation for the initial velocity is v_o = \frac{-1800g}{^4\sqrt{1+.16b^2
  • #1
Jebus_Chris
179
0

Homework Statement


A projectile has to hit a target located on the hill slope. A line of the slope is described by equation Y=.4bX. A projectile is launched at an angle of 60° with respect to the horizantal. The initial distance between the projectile and the target is 60m. Find the initial velocity of the projectile.

Homework Equations


[tex] x = v_o cos60 t [/tex]
[tex] y = v_o sin60 t - \frac{1}{2} a t^2[/tex]
[tex] a^2 + b^2 = c^2 [/tex]

The Attempt at a Solution


So we would have a triagle, hypotenuse equal to 60, base equal to x and height equal to .4bx.
So we would have...
[tex] x = v_o cos60 t [/tex]
[tex] .4bx = \sqrt {60^2 - x^2} = v_o sin60 t - \frac{1}{2} a t^2[/tex]
But then what?
 
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  • #2
In the final equation put t = x*/vo*cos(60)
So,
sqrt( 60^2 - x^2) = vo*sin(60)*x/vo*cos(60) - 1/2*g*x^2/(vo*cos60)^2

In the equation, you have two unknown quantities.
It may not be possible to solve the problem unless you know the value of b.
 
  • #3
Well, since the slope of the hill is variable then that would mean that the time it takes to reach the target isn't always the same. Therefore your answer should be in terms of b.
So first we would solve for x in terms of b.

[tex] x^2 +(.4bx)^2 = 60^2 [/tex]
[tex] x = \frac{60}{^4\sqrt{1+.16b^2}} [/tex]

We also need to solve for y in terms of b.

[tex] y = .4bx [/tex]
[tex] y = \frac{24b}{^4\sqrt{1+.16b^2}} [/tex]

So now that we know y and x in terms of b we can figure out V in terms of b.

Substitute in our x value in terms of b...
[tex] \frac{60}{^4\sqrt{1+.16b^2}} = v_o cos60t [/tex]
Solve for t...
[tex] t = \frac{60}{v_o cos60^4\sqrt{1+.16b^2}} [/tex]

Substitute our new y value in terms of b...
[tex] \frac{24b}{^4\sqrt{1+.16b^2}} = v_o sin60t - \frac{1}{2}gt^2 [/tex]
Substitute in t...
[tex] \frac{24b}{^4\sqrt{1+.16b^2}} = v_o sin60 (\frac{60}{v_o cos60 ^4\sqrt{1+.16b^2}}) - \frac{1}{2}g(\frac{60}{v_o cos60 ^4\sqrt{1+.16b^2}})^2 [/tex]
After simplifying we get [tex] v_o [/tex] to equal...
[tex] v_o = \frac{-1800g}{^4\sqrt{1+.16b^2}(24bcos^260-60sin60cos60)} [/tex]

There we go. To test this I set b=0. That would mean that it would be a flat line and x=60 and y=0 and it works out to be the right answer. [tex] v_o = 26[/tex]

BTW, i don't know how to do cubed roots or forth roots so I did ^4\sqrt. How do you actually do them?
 

FAQ: What is the initial velocity of a projectile on a sloped surface?

What is projectile motion on a slope?

Projectile motion on a slope is the movement of an object that is launched at an angle on a surface that is not horizontal. It combines the concepts of projectile motion, which involves the motion of an object through the air, and motion on an incline, which involves the motion of an object on a slanted surface.

What factors affect projectile motion on a slope?

The factors that affect projectile motion on a slope include the initial velocity of the object, the angle at which it is launched, the force of gravity, and the slope of the surface. Air resistance can also play a role, but it is often neglected in basic calculations.

How is the trajectory of a projectile on a slope different from that on a flat surface?

The trajectory of a projectile on a slope is different from that on a flat surface because gravity acts on the object in both the vertical and horizontal directions. This means that the object will follow a curved path, rather than a straight line, as it would on a flat surface.

How is the motion of a projectile on a slope calculated?

The motion of a projectile on a slope can be calculated using the principles of physics, specifically kinematics. The equations used will depend on the specific variables given, such as the initial velocity, launch angle, and slope of the surface. These equations can be solved using algebra or with the help of a computer program.

What are some real-life applications of projectile motion on a slope?

Projectile motion on a slope has many practical applications, such as in sports like skiing, snowboarding, and skateboarding. It is also important in fields like engineering and architecture, where understanding the trajectory of objects is crucial for designing structures and calculating the effects of forces on them. Additionally, projectile motion on a slope is studied in physics classrooms to illustrate the principles of motion and gravity.

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