Problem 50 - College Physics

Question

For a movie scene, a 75-kg stuntman drops from a tree onto a \(50-\mathrm{kg}\) sled that is moving on a frozen lake with a velocity of \(10 \mathrm{~m} / \mathrm{s}\) toward the shore. (a) What is the speed of the sled after the stuntman is on board? (b) If the sled hits the bank and stops, but the stuntman keeps on going, with what speed does he leave the sled? (Neglect friction.)

Step-by-Step Solution

Answer

(a) 4 m/s (b) 6.67 m/s

Step 1: Determine the Initial Momentum

The initial momentum of the system is the momentum of the sled since the stuntman is not on the sled initially. This can be calculated using the formula: \( p = m \cdot v \), where \( p \) is momentum, \( m \) is mass, and \( v \) is velocity. For the sled: \( p_{ ext{sled}} = 50 \, \text{kg} \times 10 \, \text{m/s} = 500 \, \text{kg} \cdot \text{m/s} \). The stuntman is initially not moving relative to the sled, so his initial momentum is 0.

Step 2: Calculate the Combined Mass

After the stuntman lands on the sled, the total mass of the system becomes the sum of the sled's mass and the stuntman's mass. \( m_{ ext{total}} = 50 \, \text{kg (sled)} + 75 \, \text{kg (stuntman)} = 125 \, \text{kg} \).

Step 3: Apply Conservation of Momentum to Find Final Velocity

According to the law of conservation of momentum, the total initial momentum equals the total final momentum of the system.Using the equation \( m_{ ext{sled}} \cdot v_{ ext{sled}} = m_{ ext{total}} \cdot v_{ ext{final}} \), we substitute the known values: \( 500 \, \text{kg} \cdot \text{m/s} = 125 \, \text{kg} \cdot v_{ ext{final}} \).Solving for \( v_{ ext{final}} \), we get \( v_{ ext{final}} = \frac{500}{125} = 4 \, \text{m/s} \). So, the speed of the sled after the stuntman is on board is \( 4 \, \text{m/s} \).

Step 4: Determine the Stuntman's Speed After Leaving the Sled

When the sled stops and the stuntman continues moving, the stuntman's final speed can be determined. Before hitting the bank, the system's total momentum was the same as calculated in Step 1 since momentum is conserved.After hitting the bank, the sled's momentum is 0 because it stops, but the stuntman's momentum must equal the system's momentum before stopping.Using \( p_{ ext{stuntman}} = m_{ ext{stuntman}} \times v_{ ext{stuntman}} = 75 \, \text{kg} \times v_{ ext{stuntman}} = 500 \, \text{kg} \cdot \text{m/s} \), we solve for \( v_{ ext{stuntman}} \):\( v_{ ext{stuntman}} = \frac{500 \, \text{kg} \cdot \text{m/s}}{75 \, \text{kg}} = \frac{500}{75} \approx 6.67 \, \text{m/s} \).

Key Concepts

Physics Problems

Physics problems can sometimes seem challenging, but breaking them into smaller parts makes them easier to tackle. Such problems often involve understanding how objects interact and move, governed by the laws of physics. In our exercise's context, we consider a stuntman dropping onto a moving sled. This scenario invites the application of momentum principles, a key concept in physics.

When faced with physics problems, always start by identifying the known quantities and what needs to be determined. Here, we know the masses involved and the initial velocity of the sled. We aim to determine the final velocities after various interactions. Calculating these accurately involves applying well-established physics principles, such as the conservation of momentum, which we will explore next.

Inelastic Collisions

Inelastic collisions are a fundamental concept in physics where colliding objects stick together post-collision, as opposed to elastic collisions where they bounce apart. In the context of our exercise, the stuntman's landing on the sled is an example of an inelastic collision.

During an inelastic collision:

• The two objects become a single system with combined mass.
• Kinetic energy is not conserved, but momentum is conserved.
• The objects move with the same velocity post-collision.

Understanding inelastic collisions is crucial for many real-world physics problems, such as determining final velocities after a collision. The law of conservation of momentum ensures that regardless of the collision type, the total momentum before collision is equal to the total momentum after.

Momentum Calculations

Momentum calculations are vital for solving many physics problems, including our current exercise. Momentum is calculated using the formula: \[ p = m \times v \]where \( p \) is momentum, \( m \) is mass, and \( v \) is velocity. For the sled scenario:

• Initial momentum comes solely from the sled since the stuntman is at rest initially.
• Total momentum before and after the collision is equal.

To find the final velocity after the stuntman lands on the sled, we use the combined mass of 125 kg, maintaining the initial momentum of 500 kg m/s. By solving the equation: \[ 500 = 125 \times v_{final} \]we find the sled's new speed as 4 m/s.

Moreover, once the sled stops at the bank, the stuntman continues to retain the system's momentum. Using momentum conservation again, we find his speed to be approximately 6.67 m/s after leaving the sled, providing direct insights into how momentum facilitates problem-solving in physics.