Problem 74 - College Physics

Question

A 20.0-kg toboggan with 70.0-kg driver is sliding down a frictionless chute directed 30.0° below the horizontal at 8.00 m/s when a 55.0-kg woman drops from a tree limb straight down behind the driver. If she drops through a vertical displacement of 2.00 m, what is the subsequent velocity of the toboggan immediately after impact?

Step-by-Step Solution

Answer

The subsequent velocity of the toboggan with passengers onboard immediately after the impact (based on the solution steps) can be found using the equation \(v_{f} = \frac{p_{i}}{m_{f}}\).

Step 1: Calculation of initial momentum

Calculate the initial momentum of the toboggan and the driver. Momentum is calculated by multiplying mass with velocity. Hence, the initial momentum \(p_{i}\) is given by \(p_{i} = m_{t} \cdot v_{t} + m_{d} \cdot v_{d}\), where \(m_{t} = 20.0 kg\), \(v_{t} = 8.00 m/s\), \(m_{d} = 70.0 kg\), and \(v_{d} = 8.00 m/s\). Thus, \(p_{i} = (20.0 kg + 70.0 kg) \cdot 8.00 m/s\).

Step 2: Calculation of final momentum

The final momentum equals the initial momentum since there are no external forces. Assuming the final speed of the toboggan and passengers to be \(v_{f}\), and the final mass \(m_{f} = m_{t} + m_{d} + m_{w}\), where \(m_{w} = 55.0 kg\), is the mass of the woman. Therefore, \(p_{f} = m_{f} \cdot v_{f}\). After solving the equation, we can then find the value of \(v_{f}\).

Step 3: Calculation of velocity after impact

To calculate the velocity after impact and considering that the initial momentum equals the final momentum, one can rearrange the equation to solve for \(v_{f}\) by using \(v_{f} = \frac{p_{i}}{m_{f}}\). Thus, we substitute the values found in the previous steps and solve for \(v_{f}\).

Key Concepts

Momentum

Momentum is a fundamental concept in physics that combines both mass and velocity. It allows us to analyze the movement of objects. Momentum, typically denoted by the symbol \( p \), is calculated using the formula \( p = m \cdot v \), where \( m \) is mass and \( v \) is velocity.

• Momentum is a vector quantity, meaning it has both magnitude and direction.
• The unit of momentum is kg·m/s.
• Conservation of momentum principle states that in the absence of external forces, the total momentum of a system remains constant.

In our exercise, we determine the initial momentum of the toboggan and driver by adding their masses and multiplying by the common velocity. This initial momentum is then used to find the system's velocity after the woman joins, ensuring the total momentum remains constant.

Velocity

Velocity refers to the speed of something in a given direction. It's essential when understanding motion as it not only tells us how fast an object is moving but also in which direction. This makes velocity different from speed, which only indicates how fast something is moving.

• Measured in meters per second (m/s).
• Vector quantity, meaning it has both magnitude and direction.
• Can be affected by changes in speed, direction, or both.

In the exercise, the velocity of the toboggan before and after impact is critical. Initially, the combined system moves at 8.00 m/s in a direction 30° below the horizontal. After the woman lands on the toboggan, we use the conservation of momentum to calculate the new velocity.

Mass

Mass is a measure of the amount of matter in an object. It's a fundamental property that affects how much force is needed to change an object's state of motion. The mass of an object directly influences its momentum and is a crucial part of understanding dynamics.

• Measured in kilograms (kg).
• Scalar quantity; it only has magnitude, no direction.
• Does not change when the object's location changes.

In our context, the combined masses of the toboggan, driver, and woman determine the final momentum of the system after the woman's drop. By adding each of their masses, we understand how the total system's mass affects the final velocity after the impact.