A daredevil is shot out of a cannon at 49.7◦ to the horizontal with an initial speed of 29.9 m/s. A net is positioned at a horizontal dis- tance of 49.8 m from the cannon from which the daredevil is shot.
The acceleration of gravity is 9.81 m/s2 .
At what height above the cannon’s mouth should the net be placed in order to catch the daredevil?
hf=29.9*sinTheta*time-1/2 g t^2
distance:
49.8=29.9cosTheta*time solve for time, put it in the first equation, and solve for final height, hf
To find the height at which the net should be placed, we need to consider the vertical motion of the daredevil.
First, let's break down the initial velocity into its vertical and horizontal components.
Vertical component:
vy = v * sin(θ)
where v is the initial speed and θ is the launch angle.
Horizontal component:
vx = v * cos(θ)
Take note that there are no horizontal forces acting on the daredevil (assuming no air resistance). However, there is a constant vertical force acting on the daredevil due to gravity, which will cause him to accelerate downward.
Using these components, we can analyze the vertical motion of the daredevil using the kinematic equation:
y = yo + vy*t - (1/2)*g*t^2
Here, y is the vertical displacement (height) above the cannon, yo is the initial height (which we assume to be zero), g is the acceleration due to gravity, and t is the time it takes for the daredevil to reach the net.
We need to find the time it takes for the daredevil to reach the net. Since we are given the horizontal distance (49.8 m) and the horizontal velocity (vx), we can use the equation:
x = xo + vx*t
Since the daredevil starts at xo (horizontal position = 0), we can rearrange the equation to solve for time:
t = x / vx
Now, substitute this value of t into the vertical motion equation:
y = vy * (x / vx) - (1/2) * g * (x / vx)^2
Next, we plug in the given values:
v = 29.9 m/s
θ = 49.7°
g = 9.81 m/s^2
x = 49.8 m
Using these values, we can calculate vy and vx:
vy = v * sin(θ)
vx = v * cos(θ)
Now, we substitute these values back into the equation to find the height:
y = (v * sin(θ)) * (x / (v * cos(θ))) - (1/2) * g * ((x / (v * cos(θ)))^2)
Simplifying further:
y = (sin(θ) * x) - (1/2) * (g / (v^2 * cos^2(θ)) * x^2)
Now, plug in the known values and calculate:
θ = 49.7°
x = 49.8 m
g = 9.81 m/s^2
v = 29.9 m/s
y = (sin(49.7°) * 49.8) - (1/2) * (9.81 / (29.9^2 * cos^2(49.7°)) * 49.8^2)
Calculating this expression will give you the answer for the height above the cannon's mouth where the net should be placed to catch the daredevil.