Problem 39 - 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

Impulse is 60.3 kg m/s, force is 25.66 N.

Step 1: Understand the Definition of Impulse

Impulse is defined as the change in momentum of an object when a force is applied over a specific period of time. Mathematically, it can be represented as: \[\text{Impulse} = \Delta p = m \times \Delta v\]Where \(m\) is the total mass of the system and \(\Delta v\) is the change in velocity.

Step 2: Identify the Given Values

Let's consider the total mass as the sum of the mass of the sister and the wagon: \[m = 25.0 \, \text{kg} + 8.50 \, \text{kg} = 33.50 \, \text{kg}\]The initial velocity \(v_i\) is \(0 \, \text{m/s}\) and the final velocity \(v_f\) is \(1.80 \, \text{m/s}\) after \(2.35 \, \text{s}\).

Step 3: Calculate the Change in Velocity

The change in velocity \(\Delta v\) is given by the final velocity minus the initial velocity: \[\Delta v = v_f - v_i = 1.80 \, \text{m/s} - 0 \, \text{m/s} = 1.80 \, \text{m/s}\]

Step 4: Calculate the Impulse

Using the impulse equation: \[\text{Impulse} = m \times \Delta v = 33.50 \, \text{kg} \times 1.80 \, \text{m/s} \]Calculating this gives: \[\text{Impulse} = 60.3 \, \text{kg} \, \text{m/s}\]Thus, the impulse imparted is \(60.3 \, \text{kg} \, \text{m/s}\).

Step 5: Understand the Relationship Betwen Impulse and Force

Impulse is also related to force and time through the equation: \[\text{Impulse} = F \times \Delta t\]Where \(F\) is the force applied and \(\Delta t\) is the time period over which the force is applied.

Step 6: Calculate the Force

We can rearrange the impulse equation to solve for force \(F\): \[F = \frac{\text{Impulse}}{\Delta t} = \frac{60.3 \, \text{kg} \, \text{m/s}}{2.35 \, \text{s}}\]Calculating this gives: \[F = 25.66 \, \text{N}\]Thus, the force applied on the wagon is approximately \(25.66 \, \text{N}\).

Key Concepts

Change in Momentum

When your sister sits in her wagon at rest and you start pulling her, there's an interesting change that occurs: a change in momentum. Momentum is the product of an object's mass and velocity. In this case, both the mass of your sister and the wagon combined and the change in their velocity from being at rest to moving at a speed make up the momentum change.

Initially, since the wagon is at rest, its momentum is zero. As you apply force to pull the wagon, you cause the velocity to change from zero to a certain value (in our example, 1.80 m/s). Thus, the momentum changes from zero to a new number. The total change in momentum, \( \Delta p \), is the final momentum minus the initial momentum, mathematically represented as: \[ \Delta p = m(v_f - v_i) \] where \(v_f\) and \(v_i\) are the final and initial velocities, respectively.

Force Calculation

Force is a crucial aspect when it comes to moving an object. To know how much strength you needed to move the wagon, you'll calculate the force. The force applied over a period of time contributes to the change in momentum. This relationship is represented by the impulse equation: \[ F = \frac{\text{Impulse}}{\Delta t} \]

Here, we used the determined impulse and the timeframe (2.35 seconds) to find the average force needed to achieve such motion. From this relationship, you can see how the force you exerted was effective in changing the velocity of your sister and her wagon from rest to moving.

Impulse Equation

Impulse essentially describes how the momentum of an object can be changed by the application of a force over some time. The impulse equation, \[ \text{Impulse} = \Delta p = m \times \Delta v \] is practical for calculating the impulse directly, using the product of mass and the change in velocity.

It can also provide insights into how you can achieve a particular change in momentum, either by adjusting the force applied or by changing how long the force acts. It’s a versatile tool in understanding how changes in motion occur in physics.

Velocity Change

Velocity change is the difference in velocity before and after the force has been applied. It is a direct result of applying a force that causes acceleration. You start with an initial velocity (\(v_i\)), which is often zero if the object is at rest, and end with a final velocity (\(v_f\)).

The change in velocity (\( \Delta v \)) illustrates how the speed of an object is altered. In equations, it can be written as: \[\Delta v = v_f - v_i\] This simple equation is important for calculating other values like impulse and helps in understanding the influence of force application over time.

Mass and Acceleration

Mass and acceleration are closely intertwined concepts in physics. Mass, the amount of matter in an object, influences how a force affects the object’s motion. According to Newton's second law, the acceleration (\(a\)) of an object is dependent on both the net force applied (\(F\)) and the mass (\(m\)) of the object, as laid out in the equation: \[F = m \times a\]

In this situation, both your sister and the wagon have a combined mass which is subjected to a force, leading to acceleration. As a result, this acceleration changes the velocity, exemplifying how mass plays a crucial role in determining just how much the motion of an object is altered.