Solve Physics Duck Problem: Find Min V0 & Angle for Bullet Targeting Duck

[masohman]

1. A duck flies horizontally with velocity v at a height h. When a duckhunter is right below it, he shoots with his shotgun. What is the minimum initial velocity V₀ of the bullet and the angle a at which he shoots so that the bullet hits the duck?
Obviously, given are v, h and gravity acceleration g.
2.I think I just need to take that
X_duck = vt,
X_bullet = V_0xt,
Y_bullet = V_0yt - 1/2 gt²,
V_y = V_0y - gt for the bullet.
3.I just took
X_duck = X_bullet,
Y_duck = Y_bullet,
and V_y > 0 for the bullet
but I still get a result different than the one in my book.

 

[merryjman]

All the things you have written are correct, assuming you are using h as the Y value for duck and bullet. Unless you use the Pythagorean Theorem for v_0x and v_0y then you're stuck with 2 equations and 3 unknowns (the 3rd unknown being the angle a).

 

[masohman]

Yeah, I took Y_duck = h.

My final result is that V_0min = sqrt(v² + 2gh) but the book says the answer is V_0min = sqrt(v² + 0.5gh)

 

[merryjman]

Well, I don't know why your answer is different by that factor, but I do know that this is more easily solved using conservation of energy. Since it asks for the minimum speed, you can assume the bullet's vertical velocity is zero when it hits the duck, so it need only have enough kinetic energy to be moving at the same speed as the duck:

$$\frac{1}{2}$$mv₀² = mgh + $$\frac{1}{2}$$mv²

and the correct answer for the speed comes pretty easily from that.

You can do the same thing with kinematics, I suppose, and make V_fy = 0

 

[rwisz]

I would approach this problem by first defining the variables and equations that are relevant to the situation. In this case, the variables are the initial velocity of the bullet (V0), the angle at which it is shot (a), the velocity of the duck (v), the height of the duck (h), and the acceleration due to gravity (g). The equations that can be used to solve this problem are the kinematic equations for projectile motion.

Using these equations, we can set up a system of equations to solve for V0 and a. First, we can set the horizontal distance traveled by the duck (Xduck) equal to the horizontal distance traveled by the bullet (Xbullet). This gives us the equation:

Xduck = vt = Xbullet = V0cos(a)t

Next, we can set the vertical distance traveled by the duck (Yduck) equal to the vertical distance traveled by the bullet (Ybullet). This gives us the equation:

Yduck = h + vt - 1/2gt^2 = Ybullet = V0sin(a)t - 1/2gt^2

Finally, we can set the vertical velocity of the bullet (Vy) equal to the vertical velocity of the duck (v). This gives us the equation:

Vy = v = V0sin(a) - gt

We now have a system of three equations with three unknowns (V0, a, and t). We can solve this system of equations using algebra or numerical methods to find the minimum initial velocity (V0) and angle (a) needed to hit the duck.

It is important to note that there may be multiple solutions to this problem, as there are many possible combinations of V0 and a that can result in the bullet hitting the duck. The solution in the book may be just one of these possible solutions.

In conclusion, solving this physics duck problem requires setting up and solving a system of equations using the relevant variables and equations. As a scientist, it is important to carefully define and analyze the problem before attempting to solve it.