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Chapter 7: Problem 35

A sled initially at rest has a mass of \(52.0 \mathrm{~kg}\), including all of its contents. A block with a mass of \(13.5 \mathrm{~kg}\) is ejected to the left at a speed of \(13.6 \mathrm{~m} / \mathrm{s} .\) What is the speed of the sled and the remaining contents?

Short Answer

Expert verified
Based on the conservation of momentum concept, calculate the speed of a sled with initial mass 52.0 kg after ejecting a 13.5 kg block at 13.6 m/s to the left.

Step by step solution

01

Calculate the initial momentum of the system

Since the sled is initially at rest, its initial momentum is zero. The total initial momentum of the system is also zero, as it contains only the sled and its contents.
02

Calculate the momentum of the ejected block

To calculate the momentum of the ejected block, we use the formula: Momentum = mass x velocity So, the momentum of the ejected block is: Momentum_block = \(13.5\,\mathrm{kg} \times 13.6\,\mathrm{m/s} = 183.6\,\mathrm{kg \cdot m/s}\)
03

Calculate the conservation of momentum and find the sled's momentum

Using the conservation of momentum, we know that the total momentum after the block is ejected is still zero. Therefore, the momentum of the sled and its remaining contents must be equal in magnitude (but opposite in direction) to the momentum of the ejected block: Momentum_sled = -\(183.6\,\mathrm{kg \cdot m/s}\)
04

Calculate the mass of the sled and its remaining contents

Given that the initial mass of the sled with all its contents is \(52.0\,\mathrm{kg}\), and the mass of the ejected block is \(13.5\,\mathrm{kg}\), we can calculate the mass of the sled and its remaining contents: Mass_sled = \(52.0\,\mathrm{kg} - 13.5\,\mathrm{kg} = 38.5\,\mathrm{kg}\)
05

Calculate the speed of the sled and its remaining contents

We can now determine the speed of the sled and its remaining contents using the formula: Speed = Momentum / Mass Speed_sled = \(-183.6\,\mathrm{kg \cdot m/s} / 38.5\,\mathrm{kg} = -4.76\,\mathrm{m/s}\) The negative sign indicates that the sled and its remaining contents move in the opposite direction to the ejected block. So, the speed of the sled and its remaining contents is \(4.76\,\mathrm{m/s}\) in the opposite direction to the ejected block.

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