Question

Consider two carts, of masses \(m\) and \(2 m\), at rest on a frictionless air track. If you push the lower-mass cart for \(3 \mathrm{~s}\) and then the other cart for the same length of time and with the same force, which cart undergoes the larger change in momentum? a) The cart with mass \(m\) has the larger change. b) The cart with mass \(2 m\) has the larger change. c) The change in momentum is the same for both carts. d) It is impossible to tell from the information given.

Short answer

(a) the lighter cart (b) the heavier cart (c) both have the same change in momentum Answer: (c) both have the same change in momentum

Step-by-step solution

Find the acceleration of each cart

First, use Newton's second law to find the acceleration of each cart during the push. The acceleration is given by \(a = \frac{F}{m}\), where \(F\) is the applied force and \(m\) is the mass of the object. For the cart with mass \(m\), its acceleration will be \(a_m = \frac{F}{m}\), while for the cart with mass \(2m\), its acceleration will be \(a_{2m} = \frac{F}{2m}\).

Calculate the change in velocity of each cart

The acceleration is the change in velocity over time. Use the formula \(v = a \cdot t\) to find the change in velocity for each cart, considering that the time pushed is 3 seconds for each. For the cart with mass \(m\), the change in velocity will be \(\Delta v_m = a_m \cdot t = \frac{F}{m} \cdot 3\), and for the cart with mass \(2m\), the change in velocity will be \(\Delta v_{2m} = a_{2m} \cdot t = \frac{F}{2m} \cdot 3\).

Determine the change in momentum of each cart

Now, compute the change in momentum using the formula \(\Delta p = m \Delta v\). For the cart with mass \(m\), the change in momentum will be \(\Delta p_m = m (\frac{F}{m} \cdot 3) = 3F\). For the cart with mass \(2m\), the change in momentum will be \(\Delta p_{2m} = 2m (\frac{F}{2m} \cdot 3) = 3F\).

Compare the changes in momentum

Finally, we can compare the changes in momentum of both carts. As calculated above, both carts have a change in momentum of \(3F\), which means their changes are the same. Therefore, the correct answer is option (c): The change in momentum is the same for both carts.

Key concepts

Newton's Second Law

Newton's second law of motion is a fundamental concept in physics that relates an object's mass, the force applied to it, and its acceleration. The law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This relationship is mathematically expressed as \( F = ma \), where \( F \) is the force applied, \( m \) is the object's mass, and \( a \) is the acceleration.

This principle is crucial when analyzing the motion of objects because it allows us to calculate how an object will move under the influence of a specific force. In the context of the textbook exercise, we apply Newton's second law to determine the acceleration of two different carts when the same force is applied to each. Understanding this law is key to predicting the resulting motion and, as seen in the exercise, the resulting change in momentum.

Acceleration

Acceleration refers to the rate at which an object's velocity changes over time. It can be caused by a change in speed, direction, or both. In physics, it's quantified as the change in velocity per unit of time and is expressed with the equation \( a = \frac{\Delta v}{\Delta t} \) where \( \Delta v \) is the change in velocity, and \( \Delta t \) is the time interval over which the change occurs.

In our textbook problem, the acceleration is crucial in computing the change in velocity for each cart pushed with the same force over the same time interval. A higher acceleration means a greater change in velocity, thus understanding acceleration helps us understand how quickly the motion of an object is changing due to applied forces.

Frictionless Air Track

A frictionless air track is an educational tool designed to simulate a friction-free environment. It allows physics students to study motions with minimal frictional forces impacting the results. The air track does this by blowing air through tiny holes along its surface, creating a thin cushion of air on which objects like carts can glide with almost no friction.

In the problem we are discussing, a frictionless air track ensures that no unintended forces are acting on the carts, apart from the forces we apply. This makes calculations more straightforward because it eliminates the need to account for friction, which would complicate the change in momentum computations we are concerned with.

Force and Motion

The relationship between force and motion is at the heart of classical mechanics. Force can be defined as any influence that, when applied to a body, causes it to accelerate. Motion, on the other hand, refers to the change in position of an object over time. According to Newton's laws, a force will cause an object to accelerate, which means any change in the object's velocity can be traced back to forces acting on it.

In our textbook scenario, understanding how force influences motion allows us to predict how a cart's velocity changes when a force is applied. This direct relationship makes it clear that the same force applied for the same duration will result in the same change in momentum for both carts, regardless of their masses, as confirmed by the solution to the exercise. Appreciating the connection between force and motion is essential for students to grasp how objects interact within different physical contexts.