Question

Two skaters collide and grab on to each other on frictionless ice. One of them, of mass 70.0 kg, is moving to the right at 4.00 m/s, while the other, of mass 65.0 kg, is moving to the left at 2.50 m/s. What are the magnitude and direction of the velocity of these skaters just after they collide?

Short answer

The skaters move together at 0.87 m/s to the right after the collision.

Step-by-step solution

Understanding the Conservation of Momentum

In a collision, where no external forces are acting on the system, the total momentum before the collision is equal to the total momentum after the collision, according to the law of conservation of momentum. We need to calculate the total momentum before the collision to find the velocity of the combined masses after they collide.

Calculating Initial Momentum of Each Skater

The momentum of an object is given by the product of its mass and velocity. For the first skater (70.0 kg moving at 4.00 m/s to the right):\[ p_1 = 70.0 \text{ kg} \times 4.00 \text{ m/s} = 280.0 \text{ kg m/s to the right}\]For the second skater (65.0 kg moving at 2.50 m/s to the left):\[ p_2 = 65.0 \text{ kg} \times (-2.50 \text{ m/s}) = -162.5 \text{ kg m/s to the left}\](Note: Velocity to the left is negative.)

Calculating Total Initial Momentum

The total initial momentum of the system is the sum of the momenta of the two skaters:\[ p_{\text{total, initial}} = p_1 + p_2 = 280.0 \text{ kg m/s} + (-162.5 \text{ kg m/s}) = 117.5 \text{ kg m/s}\]

Calculating the Total Mass After Collision

After the collision, the skaters grab onto each other and move together as a single object. Therefore, the total mass of the system after the collision is the sum of their masses:\[ m_{\text{total}} = 70.0 \text{ kg} + 65.0 \text{ kg} = 135.0 \text{ kg}\]

Finding the Velocity After Collision

Using the conservation of momentum, where total momentum before collision equals total momentum after collision:\[ p_{\text{total, initial}} = m_{\text{total}} \times v_{\text{final}}\]Substitute the known values:\[ 117.5 \text{ kg m/s} = 135.0 \text{ kg} \times v_{\text{final}}\]Solve for \( v_{\text{final}} \):\[ v_{\text{final}} = \frac{117.5 \text{ kg m/s}}{135.0 \text{ kg}} = 0.87 \text{ m/s}\]

Analyzing Direction of Motion

Since the total initial momentum is positive, the final direction of motion is to the right. The final velocity of the pair moving together is 0.87 m/s to the right.

Key concepts

Inelastic Collision

In physics, an inelastic collision is a type of collision where the colliding objects stick together after the impact. Unlike an elastic collision, where the objects bounce apart and total kinetic energy is conserved, inelastic collisions do not conserve kinetic energy. Instead, they conserve momentum.

• Inelastic collisions often result in deformations or energy being lost as heat or sound.
• The focus is on the conservation of the total momentum of the system.

In the context of the skaters on ice, after colliding, they hold onto each other and travel together as a single body. This behavior is a signature of a perfectly inelastic collision, demonstrating that even though energy is not conserved, we can predict outcomes using momentum conservation principles.

Momentum

Momentum is a fundamental concept in physics describing the quantity of motion an object has. It is calculated as the product of an object's mass and its velocity. Expressed mathematically, momentum $p = m imes v$, where $m$ is mass and $v$ is velocity.

• It is a vector quantity, meaning it has both magnitude and direction.
• The law of conservation of momentum states that in a closed system with no external forces, the total momentum remains constant.

For the skaters problem, each skater has its momentum calculated individually before the collision. The total system momentum is then calculated by adding these vectors, taking care of their respective directions (right is positive, left is negative). This total allows predicting the system's behavior after the collision.

Physics Problems

Solving physics problems often requires a systematic approach. Understanding the foundational principles such as momentum conservation is crucial. Let's break down the key steps you can take:

• Clearly identify what is given and what needs to be found.
• Use appropriate physical laws. For example, here we use the conservation of momentum because there are no external forces acting on the skaters.
• Translate the problem into mathematical equations, ensuring correct sign conventions for directions.
• Solve equations carefully using algebraic manipulation.

Applying these principles to the skaters, we step through calculating each skater's momentum, summing them to find total momentum, and using this to find their combined velocity after collision. This structured approach is valuable for solving a wide range of physics problems beyond collisions.