A box is at the lower end of a very slippery ramp of length (ell) that makes

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A box is at the lower end of a very slippery ramp of length \(\ell\) that makes a nonzero angle \(\theta\) with the horizontal. A worker wants to give the box a quick shove so that it reaches the top of the ramp.

(a) How fast must the box be going after the shove for it to reach its goal? (Ignore the distance traveled while the shove is executed.)

(b) What is its speed halfway up the ramp?

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