Problem 16 - College Physics

Question

You (mass 55 \(\mathrm{kg}\) ) are riding your frictionless skateboard (mass 5.0 \(\mathrm{kg}\) ) in a straight line at a speed of 4.5 \(\mathrm{m} / \mathrm{s}\) when a friend standing on a balcony above you drops a 2.5 \(\mathrm{kg}\) sack of flour straight down into your arms. (a) What is your new speed, while holding the flour sack? (b) Since the sack was dropped vertically, how can it affect your horizontal motion? Explain. (c) Suppose you now try to rid yourself of the extra weight by throwing the flour sack straight up. What will be your speed while the sack is in the air? Explain.

Step-by-Step Solution

Answer

New speed is 4.32 m/s. The sack affects only vertical motion, not horizontal. Speed remains 4.32 m/s when throwing sack up.

Step 1: Identify Known Quantities

First, list all the known quantities. The mass of the person (you) is 55 kg, the mass of the skateboard is 5 kg, the initial speed is 4.5 m/s, and the mass of the flour sack is 2.5 kg.

Step 2: Calculate Total Initial Mass

Add the mass of you and the skateboard to find the total initial mass: \( m_i = 55 \text{ kg} + 5 \text{ kg} = 60 \text{ kg} \).

Step 3: Use Conservation of Momentum

Since the system is frictionless and there are no horizontal forces, the total horizontal momentum is conserved. The initial momentum \( p_i \) is given by \( p_i = m_i \times v_i = 60 \text{ kg} \times 4.5 \text{ m/s} = 270 \text{ kg m/s} \).

Step 4: Calculate Total Final Mass

After catching the flour sack, the total final mass \( m_f \) is \( 60 \text{ kg} + 2.5 \text{ kg} = 62.5 \text{ kg} \).

Step 5: Calculate New Speed (a)

Using the conservation of momentum \( p_i = p_f \), we have \( 270 \text{ kg m/s} = 62.5 \text{ kg} \times v_f \). Solve for \( v_f \) to find the new speed: \( v_f = \frac{270}{62.5} = 4.32 \text{ m/s} \).

Step 6: Analyze Vertical Impact on Horizontal Motion (b)

Since the flour sack was dropped vertically, it does not have any horizontal momentum initially, thus it does not affect your motion horizontally. The horizontal momentum before and after catching the bag remains unchanged.

Step 7: Effect of Throwing the Sack Upwards (c)

When you throw the sack straight up, you impart only vertical momentum to the sack. The horizontal speed (4.32 m/s) remains the same while the sack is in the air because no external horizontal forces are acting on you.

Key Concepts

Newton's Laws of Motion

Newton's Laws of Motion are key principles in understanding movement and interactions between objects. They describe how forces affect motion, and they form a foundational element in physics.
Newton's First Law, often referred to as the law of inertia, states that an object will remain at rest or move at a constant velocity unless acted upon by an external force. This is clearly observed in the motion of the skateboard. Without external forces, like friction, the system’s horizontal velocity remains unchanged by the dropping of the flour sack because it doesn’t apply a horizontal force.
Newton's Second Law relates force, mass, and acceleration with the formula: \( F = ma \). In our scenario, a drastic change is not observed in horizontal motion since the force added by the vertically dropped sack is irrelevant to the horizontal path of movement.
Lastly, Newton's Third Law states that for every action, there is an equal and opposite reaction. When you throw the sack straight up, you apply a vertical force, but it doesn’t change your horizontal speed, maintaining the vessel of momentum in that direction unchanged.

Linear Momentum

Linear momentum is a concept that quantifies the amount of motion an object possesses and is calculated by the product of an object's mass and velocity. In mathematical terms, it's expressed as \( p = mv \).
The principle of conservation of momentum tells us that if no external forces are involved, the total momentum of a system remains constant. In the exercise, despite the new addition of mass (flour sack) which increased the total mass, the system's momentum is preserved because no new horizontal force was added.
When you calculate your new speed after catching the sack using the conservation formula \( m_i v_i = m_f v_f \), the result is a demonstration of how momentum remains constant, with the new speed being just a function of the new total mass.

Physics Problem Solving

Approaching physics problems methodically helps in understanding complex scenarios and finding solutions.

• Firstly, identify the given information and what is being solved.
• Secondly, understand the physical principles involved by recognizing constraints like conserved quantities (e.g., momentum).
• Break the problem into manageable steps: calculate quantities such as total mass and initial momentum.
• Apply the correct physics principles and equations, here conservation of momentum, to find new values like final speed.

This methodical approach ensures that every detail is examined and aligned with physical laws, as seen in the way the skateboarding problem is dissected into calculating masses, applying conservation of momentum, and analyzing effects of vertical action on horizontal motion.