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

Answer: The final speed of the sled with its remaining contents is 3.54 m/s.

Step-by-step solution

Calculate the total initial momentum of the system

Initial momentum is the product of mass and initial velocity. Since the sled is initially at rest, its initial velocity is 0 m/s. Thus, the initial momentum of the entire system is also 0 kg·m/s.

Calculate the final momentum of the ejected block

The final momentum of the ejected block can be calculated using the formula: momentum = mass × velocity Using the given mass of the block (13.5 kg) and its ejection speed (13.6 m/s) in the opposite direction, we get: momentum_block = 13.5 kg × (-13.6 m/s) = -183.6 kg·m/s

Use conservation of momentum to calculate the final momentum of the sled

According to the law of conservation of momentum, the total momentum of the system before and after the ejection remains constant. Therefore, the total initial momentum (0 kg·m/s) is equal to the sum of the final momenta of the sled and the ejected block. We can write this as: momentum_sled = -momentum_block Inserting the value for the final momentum of the ejected block, we have: momentum_sled = 183.6 kg·m/s

Calculate the final speed of the sled

Now, we need to find the final speed of the sled. We can use the formula for momentum: momentum_sled = mass_sled × velocity_sled We know the momentum_sled (183.6 kg·m/s) and mass_sled (52.0 kg, as mentioned in the problem). Now we can solve for the final velocity of the sled: velocity_sled = momentum_sled / mass_sled velocity_sled = 183.6 kg·m/s / 52.0 kg velocity_sled = 3.54 m/s After ejecting the block, the speed of the sled with its remaining contents is 3.54 m/s.

Key concepts

Linear Momentum

Linear momentum is a measure of the quantity of motion a body has and is a vital concept in physics, particularly in mechanics. It is a vector quantity, which means it has both magnitude and direction, and is a product of an object's mass and its velocity. Mathematically, it is represented as:
\[ \text{momentum} = \text{mass} \times \text{velocity} \]
In our exercise involving the sled and block, the concept of linear momentum is fundamental to finding the final velocity of the sled after the block is ejected. Understanding that momentum depends both on the mass and the velocity of an object helps us realize why heavier objects or those moving faster have higher momentum, and thus may require more force to change their state of motion.

Importance of Linear Momentum

Linear momentum is important because it is conserved in a closed system where there are no external forces acting, such as friction or air resistance. The conservation of linear momentum explains many physical phenomena, from the recoil of a gun when fired to the motion of objects after a collision.

Momentum-Velocity Relation

The relationship between momentum and velocity is direct and proportional, which means that as the velocity of an object increases, its momentum increases proportionally, provided the mass remains constant. The formula for momentum specifies that it is the mass of an object multiplied by its velocity:
\[ p = m \times v \]
In this case, 'p' stands for momentum, 'm' for mass, and 'v' for velocity. This is what we used in the exercise to calculate the momentum of the block, by multiplying its mass (13.5 kg) with its velocity (13.6 m/s).

Velocity's Impact on Momentum

Since velocity is a vector and has direction, a change in the direction of the object's velocity will also change the momentum's direction. This is what happened in the exercise with the block being ejected in the opposite direction, giving it a negative sign in the momentum calculation to indicate the directional change.

Law of Conservation of Momentum

The law of conservation of momentum states that in a closed system, with no external forces, the total momentum before an event must equal the total momentum after the event. This principle is crucial for solving problems involving collisions and separations.
In the textbook example, the sled and block system initially had a total momentum of zero because they were at rest. After the block is ejected, the principle of conservation of momentum tells us that the sled must move in the opposite direction to conserve momentum. We used this law to equate the final momentum of the sled to the negative of the momentum of the ejected block, thereby allowing us to solve for the sled's final speed.

Applications of Conservation of Momentum

• Spacecraft propulsion: Astronauts use the conservation of momentum when they throw objects in the opposite direction to move the spacecraft.
• Sports: In games like pool or billiards, striking a ball leads to the transfer and conservation of momentum between the balls.
• Traffic accidents: Investigators use the conservation of momentum to reconstruct the events of vehicular collisions.

The consistency of the law of conservation of momentum makes it a powerful tool for physicists and engineers. By understanding this law, students can solve a wide range of problems in mechanics.