A gunman standing on a sloping ground fires up the slope. The initial speed of the bullet is v0= 340 m/s. The slope has an angle α= 12 degrees from the horizontal, and the gun points at an angle θ from the horizontal. The gravitational acceleration is g=10 m/s2.
(a) For what value of θ ( where θ>α) does the gun have a maximal range along the slope? (in degrees, from the horizontal)
θ=
unanswered
(b) What is the maximal range of the gun, lmax, along the slope? (in meters)
lmax=
a) 45+a/2=51 degrees
b)(1−sin)*v0^2/(g*cos^2(á))
@Greco b one more time
Lordy me, and I did this one by mistake yesterday not knowing it was an 8:01 final question
(v_0^2*(1-sin(alpha))/(g*cos^2(alpha))
To find the value of θ that corresponds to the maximal range along the slope, we need to analyze the projectile motion of the bullet. Here's how to do it:
Step 1: Resolve the initial velocity into horizontal and vertical components. Since the gun points at an angle θ from the horizontal, the horizontal component of the velocity will be v0x = v0 * cos(θ), and the vertical component will be v0y = v0 * sin(θ).
Step 2: Express the bullet's time of flight in terms of the launch angle θ. The time of flight t can be calculated using the equation t = 2*v0y / g.
Step 3: Calculate the horizontal range R along the slope using the equation R = v0x * t.
Step 4: Differentiate R with respect to θ and set the derivative equal to zero to find the angle that corresponds to maximum range. This will give you the value of θ for maximum range.
Now let's solve the problem:
(a) Find the value of θ for maximum range:
- Resolve the initial velocity into components:
v0x = v0 * cos(θ)
v0y = v0 * sin(θ)
- Calculate the time of flight:
t = 2 * v0y / g
- Calculate the horizontal range:
R = v0x * t
- Differentiate R with respect to θ:
dR/dθ = (-v0 * sin(θ)) * t
- Set dR/dθ equal to zero and solve for θ:
0 = (-v0 * sin(θ)) * t
0 = -v0 * sin(θ) * (2 * v0 * sin(θ) / g)
0 = -2 * v0^2 * sin^2(θ) / g
Since θ > α, the angle cannot be 0 degrees.
Thus sin^2(θ) = 1, and the equation simplifies to:
0 = -2 * v0^2 / g
0 = -2 * (340 m/s)^2 / 10 m/s^2
0 = -2 * 115600 / 10
0 = -23120
This equation doesn't have a real solution. Hence, there is no value of θ greater than α that will give the gun a maximal range along the slope.
Therefore, θ is undefined.
(b) As there is no maximum range along the slope, the maximal range (lmax) is also undefined.