Passing Through a Sphere

Classical Mechanics Level 4

Consider a spherical region with radius $1 \, \text{m}$ centered on $(x,y,z) = (2,3,5) \, \text{m}$ . A projectile is launched from the origin with a speed of $20 \, \text{m/s}$ . The projectile enters the sphere and then leaves it at $(x,y,z) = (2,3,6) \, \text{m}$ .

If gravity is $10 \, \text{m/s}^2$ in the negative- $z$ direction, how much time (in seconds) does the projectile spend within the sphere?

The answer is 0.09737.

1 solution

Otto Bretscher
Dec 1, 2018

Since the projectile moves in the plane $3x=2y$ , it is convenient to introduce a $u,v$ coordinate system with orthonormal basis $\frac{1}{\sqrt{13}}(2,3,0),(0,0,1)$ . The path of the projectile is of the form $u=20(\cos\theta) t, v=20 (\sin\theta) t-5t^2$ . The smaller solution of $u=\sqrt{13},v=6$ is $t\approx 0.3817,\theta=1.0789$ . The other solution of $(u-\sqrt{13})^2+(v-5)^2=1$ is $t\approx 0.2843$ , so that the answer is $\Delta t\approx 0.0974$ .

Passing Through a Sphere

The answer is 0.09737.

1 solution

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