A daredevil is shot out of a cannon at 41.6
◦
to the horizontal with an initial speed of
28.6 m/s. A net is positioned a horizontal
distance of 29 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 answer this question, we need to determine the height above the cannon at which the net should be placed in order to catch the daredevil. We can do this by using the equations of motion for projectile motion.
First, let's break the given information into the knowns and unknowns:
Knowns:
- Initial horizontal speed (Vx) = 28.6 m/s
- Horizontal distance from cannon to net (x) = 29 m
- Acceleration due to gravity (g) = 9.8 m/s^2
Unknown:
- Height above the cannon (y)
Now, let's analyze the motion in the horizontal and vertical directions separately.
Horizontal Motion:
The horizontal motion of the daredevil is uniformly at a constant speed, so there is no acceleration in the horizontal direction. Therefore, we can use the formula:
x = Vx * t
where x is the horizontal distance (29 m in this case), Vx is the initial horizontal speed (28.6 m/s in this case), and t is the time of flight.
Vertical Motion:
In the vertical direction, the daredevil experiences uniform acceleration due to gravity. We can use the following equation to find the time of flight:
y = Vy0 * t - (1/2) * g * t^2
where y is the vertical displacement, Vy0 is the initial vertical speed (which is 0 as the daredevil is shot upwards), g is the acceleration due to gravity (9.8 m/s^2 in this case), and t is the time of flight.
Now, let's find the time of flight (t) using the horizontal motion equation:
29 m = 28.6 m/s * t
Solving this equation for t:
t = 29 m / 28.6 m/s
≈ 1.0139 s
Now that we have the time of flight, we can substitute it into the vertical motion equation to find the height above the cannon (y):
y = 0 * 1.0139 s - (1/2) * 9.8 m/s^2 * (1.0139 s)^2
Simplifying this equation:
y = -4.9 m/s^2 * (1.028 s)^2
≈ -4.9 m/s^2 * 1.057 s^2
≈ -5.1833 m^2/s^2
Since the height above the cannon cannot be negative, we can discard the negative sign and take the absolute value of the result:
y ≈ 5.1833 m
Therefore, the net should be placed at a height of approximately 5.1833 meters above the cannon in order to catch the daredevil.