Question

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

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

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.

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 = \(13.5\,\mathrm{kg} \times 13.6\,\mathrm{m/s} = 183.6\,\mathrm{kg \cdot m/s}\)

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 = -\(183.6\,\mathrm{kg \cdot m/s}\)

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 = \(52.0\,\mathrm{kg} - 13.5\,\mathrm{kg} = 38.5\,\mathrm{kg}\)

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.

Key concepts

Momentum Calculation

Momentum is a fundamental concept in physics that describes the quantity of motion an object possesses. It is calculated as the product of an object's mass and velocity. For any object in motion, momentum is given by the formula: \[\text{Momentum} = \text{mass} \times \text{velocity}\]In the example provided, an ejected block has a mass of 13.5 kg and a velocity of 13.6 m/s. By applying the formula, we calculate its momentum as:\[\text{Momentum}_{\text{block}} = 13.5\,\mathrm{kg} \times 13.6\,\mathrm{m/s} = 183.6\,\mathrm{kg \cdot m/s}\]This calculation helps us understand how much motion the block carries as it separates from the sled. The principle of conservation of momentum tells us that this motion will influence the sled as well.

Motion and Direction

In physics, motion doesn't just involve speed; direction plays a crucial role as well. When dealing with conservation of momentum, direction is integral because momentum is a vector quantity. It has both magnitude and direction. In our scenario, the block is ejected to the left. Therefore, the sled and its remaining contents must move in the opposite direction to maintain the system's total momentum at zero. Initially, the entire system is at rest, meaning there is no net movement. After the block is ejected, it carries momentum to the left. Consequently, the sled must move to the right with an equal magnitude of momentum but in the opposite direction: - The calculated momentum of the block is 183.6 kg·m/s to the left. - The sled carries -183.6 kg·m/s, indicating motion to the right. This ensures that the system's total momentum remains balanced.

Physics Problem Solving

Solving physics problems often requires a step-by-step approach, especially when applying concepts like conservation of momentum. Let's dissect how this approach is applied in this problem involving a sled and a block.First, assess the initial conditions: The entire system (sled and block) starts at rest, meaning the initial total momentum is zero. Next, when the block is ejected, identify its movement and calculate the momentum using its mass and velocity.Once the block is in motion, apply the conservation of momentum. Per this principle, the total momentum of a closed system remains constant, assuming no external forces act upon it. In this case, the sled must move in the opposite direction to the block to conserve momentum.Finally, compute the speed of the sled using the calculated momentum and remaining mass. By dividing the sled's momentum by its remaining mass, calculate the sled's velocity. Be sure to consider direction, seeing as the resulting negative value reflects the sled's opposite movement:\[\text{Speed}_{\text{sled}} = \frac{\text{Momentum}_{\text{sled}}}{\text{Mass}_{\text{sled}}} = \frac{-183.6\,\mathrm{kg \cdot m/s}}{38.5\,\mathrm{kg}} = -4.76\,\mathrm{m/s}\]Through careful analysis and strategic application, you can solve complex physics problems effectively.