(Projectile Motion) You are being asked to bombard an enemy fortress that sits on a 40 m tall cliff 100 m from your location. Your cannon has an elevation of 37 degrees. What velocity must your cannonball have to hit the walls of the fortress?
First check if it is feasible!
The fortress is 40 m high and 100 m away.
The angle θ=tan-1(40,100)=21.8°.
Since θ < angle of elevation of cannon of 37°, it is feasible.
Let the muzzle velocity be v0, and the angle of elevation be φ.
Horizontal component of velocity=v0 cos(φ)
Vertical component of velocity = v0 sin(φ)
Let Sh(t)=horizontal distance at time t after firing, and
Sv(t)=vertical distance at time t.
g=acceleration due to gravity = 9.8 m/s²
Then
Sh(t)=v0 cos(φ) t
To hit target, we get
Sh(t)=100 m = v0 cos(φ) t
t=100/(v0cos(φ)) ....... (1)
Solve for t in terms of v0 and φ.
Sv(t) = 40 = v0 sin(φ) t - (1/2)gt²
Substitute t into above equation to and solve for v0.
There should be two solutions. Any negative value of v0 should be rejected.
Each retained solution must be substituted into equation (1). Any solution which results in a negative value of t should also be rejected.
I get approximately v0=47 m/s and t between 2.5 to 3 seconds.
thank you very much.. i understand what you are saying, but can you show me the algebra steps to get the 2 solutions?
i keep substituting the first equation into the second and try to work it out, but for some reason i am still doing it wrong.
It's time you show me your work!
I will be pleased to help you find the problem if there is any.
my variables are a little different than yours, but here is what i have..
Yf=Vosinθ (tf)-.5g(tf)²
Yf=Vosinθ(xf)/Vocosθ-.5g(xf/Vocosθ)²
i'm having trouble isolating Vo. am i supposed to use the quadratic equation? my algebra is not very good.. so if you can point me in the right direction it would help alot.
Fortunately there is no quadratic equation to solve, just a square-root.
Yf=Vosinθ(xf)/Vocosθ-.5g(xf/Vocosθ)²
Yf- sinθ(xf)/ cosθ=.5g(xf/Vocosθ)²
Vo²
=0.5g(xf/cosθ)²/(Yf-(xf)tanθ), or
Vo=sqrt(0.5g(xf/cosθ)²/(Yf-(xf)tanθ))
Now take out your calculator and substitute the values. You should get about 46.6 m/s.
ahhh i finally got it!!! thank you so much!
How much did you get?
What velocity did you get?
To determine the velocity required for the cannonball to hit the walls of the fortress, we can break down the problem into two components: horizontal and vertical motion.
First, let's analyze the horizontal motion. The cannonball will be affected by gravity in the vertical direction but will have a constant velocity in the horizontal direction.
Using the given information, the horizontal distance between your location and the fortress is 100 m. Therefore, the horizontal velocity (Vx) remains constant throughout the motion.
Next, let's examine the vertical motion. The cannonball will be acted upon by gravity in the vertical direction. We know that the fortress is located on a 40 m tall cliff. Additionally, the cannon has an elevation of 37 degrees.
To solve this problem, we can split the initial velocity (V_0) of the cannonball into horizontal (Vx) and vertical (Vy) components using trigonometry.
Vx = V_0 * cos(37°)
Vy = V_0 * sin(37°)
The time of flight (t) can be calculated using the vertical motion:
Y = Y_0 + V_0y * t - (1/2) * g * t^2
where Y is the vertical distance, Y_0 is the initial height, V_0y is the initial vertical velocity, t is the time of flight, and g is the acceleration due to gravity.
Given:
Y = 40 m
Y_0 = 0 (as we take the initial height as the reference point)
V_0y = V_0 * sin(37°)
g = 9.8 m/s^2 (approximate value)
Using this information, we can rearrange the equation to solve for t:
t = [V_0y ± sqrt((V_0y)^2 + 2gY)] / g
Since we want the time it takes for the cannonball to hit the fortress, we can choose the positive square root.
Now, we can use the horizontal motion to find the time of flight as:
t = d / Vx
where d is the horizontal distance.
Setting these two equations equal to each other:
[V_0y ± sqrt((V_0y)^2 + 2gY)] / g = d / Vx
Substituting the values we know:
[V_0 * sin(37°) ± sqrt((V_0 * sin(37°))^2 + 2 * 9.8 * 40)] / 9.8 = 100 / (V_0 * cos(37°))
Rearrange the equation to isolate V_0:
V_0 = sqrt((100 * 9.8 * cos(37°))^2 / (100 * sin(37°) + 40 * g))
Now you can calculate the value of V_0 using this equation.
Please note that this is a simplified explanation, and various factors like air resistance and projectile shape are not considered in this analysis.