Problem 42 - College Physics

Question

Your little sister (mass 25.0 \(\mathrm{kg}\) ) is sitting in her little red wagon (mass 8.50 \(\mathrm{kg} )\) at rest. You begin pulling her forward and continue accelerating her with a constant force for 2.35 \(\mathrm{s}\) at the end of which time she's moving at a speed of 1.80 \(\mathrm{m} / \mathrm{s}\) . (a) Calculate the impulse you imparted to the wagon and its passenger. (b) With what force did you pull on the wagon?

Step-by-Step Solution

Answer

(a) The impulse is 60.3 kg·m/s. (b) The force exerted is 25.66 N.

Step 1: Understanding Impulse

Impulse is the product of the change in momentum of an object and the time during which the force is applied. It is given by the formula: \( J = F \cdot \Delta t = \Delta p \), where \( J \) is the impulse, \( F \) is the force, and \( \Delta t \) is the time duration. The change in momentum \( \Delta p \) is calculated as \( m \cdot v_f - m \cdot v_i \), with \( m \) as mass, \( v_f \) as final velocity, and \( v_i \) as initial velocity.

Step 2: Calculate Total Mass

Determine the total mass of the system, which includes both your sister and the wagon:\[ m = m_{sister} + m_{wagon} = 25.0 \; \mathrm{kg} + 8.50 \; \mathrm{kg} = 33.5 \; \mathrm{kg} \]

Step 3: Initial and Final Momentums

Since the system starts from rest, the initial velocity \( v_i \) is 0. The final velocity \( v_f \) is given as 1.80 m/s. Therefore: \[ \Delta p = m \cdot v_f - m \cdot v_i = 33.5 \; \mathrm{kg} \cdot 1.80 \; \mathrm{m/s} - 33.5 \; \mathrm{kg} \cdot 0 = 60.3 \; \mathrm{kg \cdot m/s} \]

Step 4: Calculate the Impulse

The impulse is equal to the change in momentum. Thus, the impulse imparted to the wagon and passenger is: \( J = 60.3 \; \mathrm{kg \cdot m/s} \).

Step 5: Calculate the Force

To find the force, use the impulse formula related to force and time: \( J = F \cdot \Delta t \). Re-arranging for \( F \), we have \( F = \frac{J}{\Delta t} \). Substituting the known values: \[ F = \frac{60.3 \; \mathrm{kg \cdot m/s}}{2.35 \; \mathrm{s}} = 25.66 \; \mathrm{N} \]

Key Concepts

Newton's laws of motion

Newton's laws of motion are fundamental when studying mechanics, providing a basis for understanding how objects move. Newton's first law states that an object will remain at rest, or in uniform motion unless acted upon by an external force. This introduces the idea of inertia and helps explain how motion requires a force to change an object's velocity.
The second law, which is deeply connected to the problem at hand, relates force, mass, and acceleration. It states that the force acting on an object is proportional to the mass of the object and the acceleration it experiences: \( F = m \times a \). This law is crucial for problems involving moving objects since it allows us to calculate the force required to achieve a certain acceleration.
Finally, Newton's third law tells us that for every action, there is an equal and opposite reaction. Although we are primarily focusing on the first two laws in our exercise, understanding these principles as a whole offers a comprehensive view of why forces affect motion as they do.

Calculation of force

The calculation of force is an application of Newton's second law. From the law, we know that force is the product of mass and acceleration: \( F = m \times a \). In our exercise, we discovered the total force exerted by using the impulse formula, which interconnects force with change in momentum over a given time.
In practice, to determine the exact force you applied to the wagon, we used the impulse calculated earlier: \( J = 60.3 \, \mathrm{kg \cdot m/s} \). Impulse is defined as \( J = F \cdot \Delta t \), where \( \Delta t \) is the time during which the force was applied, calculated as \( 2.35 \mathrm{\,s} \) in this problem.
Solving for force, we rearrange the formula to: \( F = \frac{J}{\Delta t} \). By substituting \( J = 60.3 \, \mathrm{kg \cdot m/s} \) and \( \Delta t = 2.35 \mathrm{\,s} \), the force is found to be \( 25.66 \, \mathrm{N} \). This is how much force you must apply to keep the wagon accelerating at the given rate.

Change in velocity

In our exercise, change in velocity is a key component when determining momentum and impulse. Velocity is a vector quantity, meaning it has both magnitude and direction, and any change in these aspects constitutes a change in velocity.
Initially, your sister and her wagon were at rest, meaning their initial velocity \( v_i \) was \( 0 \mathrm{\,m/s} \). After applying force, their velocity changed to a final speed of \( 1.80 \mathrm{\,m/s} \). The change in velocity, therefore, is \( v_f - v_i = 1.80 \mathrm{\,m/s} - 0 = 1.80 \mathrm{\,m/s} \). This information is fundamental in calculating the change in momentum because momentum change involves the mass of the objects and the change in velocity according to \( \Delta p = m \times (v_f - v_i) \).
This step is essential in bridging the motion with the force applied, making it possible to determine both impulse and resultant force using Newton's laws of motion. Understanding these changes can be exciting, especially when you see how small pushes translate into significant movements.

Mass and acceleration

Mass and acceleration play crucial roles in understanding Newton's second law of motion. The mass of an object is a measure of its inertia – how much resistance an object has to changes in its state of motion.
In the problem, the total mass includes both your sister and her wagon, calculated as \( 25.0 \, \mathrm{kg} + 8.50 \, \mathrm{kg} = 33.5 \, \mathrm{kg} \). This mass is essential when computing force, as force is directly proportional to both mass and acceleration (\( F = m \times a \)). Thus, without knowing the total mass, we can't accurately calculate the force needed for the given acceleration.
Acceleration is defined as the rate of change of velocity over time. When you apply force to the wagon, you're causing it to accelerate, changing its speed over the particular time frame of \( 2.35 \, \mathrm{s} \). Understanding how to manipulate these quantities is key in physics, demonstrating the robust connection between mass, force, and acceleration in every motion-related scenario.