On Friday in class, we learned that if you were to aim right at the bulls-eye, the arrow would fall under gravity and drop below the intended target. If you intend to hit a bulls-eye it is necessary that you aim slightly above it. Suppose the arrow speed is the same (70 m/s) and the target is 10 meters away. What launch angle, Θ, is needed so that the arrow hits the bulls-eye?
V=70m/s
hf=hi+70sinTheta*t-4.9t^2
but horizontally
10=70cosTheta*t
in the horizontal equation
t= 10/70costheta
put that into the vertical first equation, and solve for theta.
hf=hi
while john is traveling along an interstate highway, he notice a 160-mile marker a he passes through town. later john passes another mile marker, 115.
what is the distance between town and john's current location? what is his current position?
To find the launch angle (Θ) that is needed for the arrow to hit the bulls-eye, we can use projectile motion equations and the knowledge that the arrow will drop below the intended target due to the force of gravity.
First, let's break down the initial velocity (V) into its horizontal and vertical components. The horizontal component will remain constant and equal to V throughout the motion, while the vertical component will change due to the influence of gravity.
So, the horizontal component of the velocity, Vx, is given by:
Vx = V * cos(Θ)
And the vertical component of the velocity, Vy, is given by:
Vy = V * sin(Θ)
Since the target is at a distance of 10 meters, we need to find the time it takes for the arrow to reach the target. We can use the horizontal component of velocity to calculate this.
The time, t, is given by:
t = distance / horizontal velocity
Substituting the given values:
t = 10 / Vx
Now, let's find the vertical displacement, which is the difference between the target's height and the arrow's height.
The vertical displacement, h, is given by:
h = Vy * t + 0.5 * g * t^2
Where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Since we want the arrow to hit the bulls-eye, the vertical displacement should be zero. Therefore, we can set the equation equal to zero and solve for the launch angle, Θ.
0 = Vy * t + 0.5 * g * t^2
Substituting the expressions for Vy and t:
0 = V * sin(Θ) * (10 / Vx) + 0.5 * g * (10 / Vx)^2
Simplifying:
0 = sin(Θ) * (10 / cos(Θ)) + 0.5 * g * (10 / cos(Θ))^2
Now, we can solve this equation to find the launch angle (Θ) using numerical methods or calculators capable of solving equations.
Note: The above equation can be simplified further by substituting Vx = V * cos(Θ) into the equation.
Once the equation is solved, the obtained launch angle (Θ) will be the value needed to hit the bulls-eye.