Question

A \(423.5-\mathrm{N}\) force accelerates a go-cart and its driver from \(10.4 \mathrm{~m} / \mathrm{s}\) to \(17.9 \mathrm{~m} / \mathrm{s}\) in \(5.00 \mathrm{~s}\). What is the mass of the go-cart plus driver?

Short answer

Answer: The mass of the go-cart and its driver is approximately 56.47 kg.

Step-by-step solution

Identify the given values and the unknown value we need to find

Given values: - Force (F) = 423.5 N - Initial velocity (v_initial) = 10.4 m/s - Final velocity (v_final) = 17.9 m/s - Time (t) = 5.00 s Unknown value: - Mass (m)

Calculate the acceleration using the initial and final velocities and the time

Acceleration (a) can be calculated using the formula: a = (v_final - v_initial) / t a = (17.9 m/s - 10.4 m/s) / 5.00 s a = 7.5 m/s²

Use Newton's second law of motion to find the mass

Newton's second law states that: F = m × a We need to find the mass (m), so we will rearrange the formula to find m: m = F / a Now, we plug in the given values and the calculated acceleration: m = 423.5 N / 7.5 m/s² m = 56.47 kg The mass of the go-cart plus driver is approximately 56.47 kg.

Key concepts

Understanding Force and Acceleration

When exploring the dynamic relationship between force and acceleration, we delve into one of the most fundamental principles of classical mechanics, beautifully captured by Newton's second law of motion. This law provides the mathematical equation that connects force (\( F \)), mass (\( m \)), and acceleration (\( a \) as shown \( F = m \times a \). In the context of our exercise, a known force propels a go-cart and its driver forward, resulting in a change of speed.

To get a deep understanding, consider force as the push or pull acting upon an object, which can cause the object to speed up, slow down, or change direction. Now, acceleration is the rate at which an object changes its velocity. Therefore, the greater the force applied to an object, the greater the acceleration, assuming the mass remains constant. In practice, if you push a toy car gently, it will move slowly; push it harder, and it accelerates more quickly. It's this simple concept that underpins much of motion in physics.

An equally crucial point is that acceleration is directly proportional to the force applied and inversely proportional to the mass of the object. What this means is that a heavier object will require more force to achieve the same acceleration as a lighter one. Hence, understanding this relationship is critical when we attempt to calculate the mass of an object based on the force applied and the acceleration observed, as demonstrated in our original exercise.

The Art of Mass Calculation

Mass calculation in physics is essentially an exercise in detective work where we use the clues provided by the laws of motion to determine the 'heaviness' of an object. Mass is a measure of the amount of matter in an object, commonly perceived as its weight (though weight is technically mass under the pull of gravity). One might say mass is how we quantify inertia, the resistance of any physical object to any change in its velocity, including changes to the object's speed, or direction of motion.

In the example provided, the mass is not directly observable, but its effects are evident through the force exerted and the corresponding acceleration. Here are the fundamental steps for mass calculation in our scenario:

• Recognize Newton's second law equation (\( F = m \times a \) where the mass (\( m \) is sought.
• Isolate mass on one side of the equation to solve for it, thus becoming \( m = F / a \).
• Insert the known values of force and acceleration into the equation to calculate the mass.

A critical insight for students is that mass remains constant, regardless of the place and the condition (ignoring relativistic speeds and extreme gravitational forces), while the weight varies with the local strength of gravity. This distinction is crucial for solving such physics problems.

Change in Velocity Explained

Velocity is a vector quantity, which means it encompasses both speed — the rate at which an object covers distance — and direction of travel. Change in velocity, therefore, can be due to a change in the object's speed, its direction, or both. Moreover, velocity is not just about how fast an object is moving; it's also about where it's going.

In our exercise, we calculate acceleration by using the change in velocity over time. This is encapsulated in the formula \( a = (v_{final} - v_{initial}) / t \). Simply put, if a car accelerates from rest to 60 miles per hour northward, its change in velocity is 60 mph north. But if it then decelerates to a stop, the change in velocity is 60 mph south (the opposite direction).

Here's how it breaks down in the context of our exercise - the 'change' part is the difference between the go-cart's final velocity and its initial velocity. By dividing this change by the time in which the change occurred, acceleration is obtained which proved crucial for solving the mass calculation problem in the step by step solution.

Understanding change in velocity is fundamental, not only because it appears so often in equations, but because it's a tangible concept that connects the abstract world of physics with the real motions we observe every day. It's the palpable rush of speed one feels when a car accelerates, or the lurching sensation as the same car comes to a quick stop.