A daredevil is shot out of a cannon at 54 degrees to the horizontal with an initial speed of 33.3 m/s. A net is positioned a horizontal distance of 41.8 m from the cannon. At what height above the cannon should the net be placed in order to catch the daredevil?
The acceleration of gravity is 9.8 m/s^2.
Answer in units of m.
To find the height above the cannon where the net should be placed, we can use the projectile motion equations.
First, let's break down the initial velocity of the daredevil into its x and y components. The horizontal component is given by Vx = v0 * cos(theta), and the vertical component is given by Vy = v0 * sin(theta).
Given:
Initial speed, v0 = 33.3 m/s
Launch angle, theta = 54 degrees
So, Vx = 33.3 * cos(54) ≈ 19.564 m/s
And, Vy = 33.3 * sin(54) ≈ 26.244 m/s
Next, let's find the time of flight (t), which is the time it takes for the daredevil to reach the net. We can use the vertical component of the motion since the height is determined by the vertical motion.
Using the equation y = Vyi * t + (1/2) * (-g) * t^2, and knowing that at the highest point Vyf = 0, we can rearrange it as:
0 = Vy - g * t
t = Vy / g
Substituting the known values, t = 26.244 / 9.8 ≈ 2.677 seconds
Now, we can find the height (h) above the cannon where the net should be placed. We can use the equation y = Vyi * t + (1/2) * (-g) * t^2 with the calculated time.
y = Vy * t + (1/2) * (-g) * t^2
y = 26.244 * 2.677 + (1/2) * (-9.8) * (2.677)^2
Calculating the expression gives:
y ≈ 70.06 m
Therefore, the net should be placed approximately 70.06 meters above the cannon in order to catch the daredevil.
To find the height above the cannon at which the net should be placed, we can use the equations of motion and basic kinematics.
Step 1: Resolve the initial velocity into horizontal and vertical components.
Given:
Initial speed, v₀ = 33.3 m/s
Angle of projection, θ = 54 degrees
The horizontal component of velocity can be found using:
v₀x = v₀ * cos(θ)
The vertical component of velocity can be found using:
v₀y = v₀ * sin(θ)
Step 2: Determine the time of flight.
The time of flight, t, can be calculated using the vertical component of velocity and the acceleration due to gravity:
t = (2 * v₀y) / g
where g = 9.8 m/s^2.
Step 3: Calculate the horizontal distance traveled by the daredevil.
The horizontal distance traveled can be found using the equation:
dx = v₀x * t
Step 4: Calculate the maximum height reached by the daredevil.
The maximum height reached can be found using the equation:
h = (v₀y^2) / (2 * g)
Step 5: Find the height above the cannon for the net placement.
The height above the cannon should be equal to the maximum height reached by the daredevil plus the height of the net:
h_net = h + height of net
Now, let's calculate the answer.
Using the above formulas and the given values:
v₀x = 33.3 * cos(54)
v₀y = 33.3 * sin(54)
t = (2 * v₀y) / g
dx = v₀x * t
h = (v₀y^2) / (2 * g)
Substitute these values into the formulas and solve for h_net:
h = ((33.3 * sin(54))^2) / (2 * 9.8) (Substituting the values)
h = 59.63 m (Calculating the value of h)
Now, add the height of the net above the maximum height reached:
h_net = 59.63 + height of net
The height of the net above the cannon will depend on the specific requirements or constraints for catching the daredevil.
The horizontal velocity component is 33.3 cos54 = 19.57 m/s
The daredevil reaches the 41.8 m distance in T = 41.8/19.57 = 2.14 seconds
His height above the ground at that time can be calculated using
Y = 33.3 sin54*T - (g/2)*T^2.