During World War I, the Germans had a gun
called Big Bertha that was used to shell Paris.
The shell had an initial speed of 2.65 km/s at
an initial inclination of 41.9◦ to the horizontal.
The acceleration of gravity is 9.8 m/s
2
.
How far away did the shell hit?
Answer in units of km
the initial vertical velocity is __ (2650 m/s) * cos(41.9º)
the time of flight is the initial vertical velocity divided by the gravitational acceleration __ 2 * [(2650 m/s) * sin(41.9º)] / (9.8 m/s^2)
the horizontal velocity is __ (2.65 km/s) * cos(41.9º)
the range is the horizontal velocity multiplied by the flight time
To find the distance at which the shell hit, we can use the kinematic equation that relates distance, initial velocity, angle of inclination, and acceleration due to gravity.
The equation we can use is:
d = (v^2 * sin(2θ)) / g
Where:
d is the distance traveled by the shell
v is the initial speed of the shell
θ is the angle of inclination
g is the acceleration due to gravity
First, let's convert the initial speed from km/s to m/s:
v = 2.65 km/s * 1000 m/km = 2650 m/s
Next, let's convert the angle of inclination from degrees to radians:
θ = 41.9° * π/180 = 0.731 radians
Now, we can substitute these values into the equation to find the distance:
d = (2650^2 * sin(2 * 0.731)) / 9.8
Evaluating this expression, we get:
d ≈ 1531766 m
Finally, let's convert the distance from meters to kilometers:
d = 1531766 m / 1000 m/km = 1531.766 km
Therefore, the shell hit at a distance of approximately 1531.766 km.