The movie Elf exemplifies all things that are great about winter and the holiday season. One of the most impressive scenes is when Buddy throws a snowball across the park to knock over the last remaining attacker after he and Michael are ambushed on their way home from school.
I was always curious about how a small snowball could have caused the boy to be launched into the air causing a major wipe-out. To begin the analysis, I first looked back to kinematics to calculate the velocity of the snowball when it hits the boy:
From watching the clip I assumed these values:
Vo = ?
θ = 45o
Vox = (cos45o) Vo
Voy = (sin45o) Vo
Δx = 45 m
Δy = 0 m
t = 3.0 s
ax = 0 m/s2
ay = -9.8 m/s2
I then calculated the initial velocity in the x-direction using kinematic equations:
Δx = (Vox)t + 1/2at2
45 m = (Vox)(3.0 s)
Vox = 15 m/s
Since (cos45o) = (sin45o), the initial velocity in the x-direction is the same as the initial velocity in the y-direction. We then can calculate the initial totally velocity, which will equate to the final velocity since there is no change in height and there is no acceleration in the x-direction.
Vo = √[(Vox )2 + (Voy)2]
Vo = √[(15 m/s)2 + (15 m/s)2]
Vo = 21.21 m/s
I then used conservation of momentum to determine how the snowball would affect the boy:
m1Bv1B + m1Sv1S = m2Bv2B + m2Sv2S
(60 kg)(2.24 m/s) + (0.25 kg)(21.21 m/s) = (60 kg)(v2B)
v2B = 2.33 m/s
Assuming the collision between the snowball and the boy was elastic, the boy's velocity would only increase by 0.09 m/s. This would logically not cause the boy to be launched into the snow. We can also consider this problem qualitatively with torque. The snowball hits the boy approximately half way up his back. If we assign the axis of rotation to be horizontal through the center of the boy's body, a force that acts at or very close to that point will not exert a torque on the boy. Therefore, the snowball that hits in the center of his back will not cause him to flip face-down into the snow.
Although this scene was exciting and impressive in the movie, it over-dramatizes the effects of being hit with a snowball and gives us another example of bad physics in movies.