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Chapter 9: Problem 10

A \(1000 \mathrm{~kg}\) automobile is at rest at a traffic signal. At the instant the light turns green, the automobile starts to move with a constant acceleration of \(4.0 \mathrm{~m} / \mathrm{s}^{2} .\) At the same instant a \(2000 \mathrm{~kg}\) truck, traveling at a constant speed of \(8.0 \mathrm{~m} / \mathrm{s}\), overtakes and passes the automobile. (a) How far is the com of the automobile-truck system from the traffic light at \(t=3.0 \mathrm{~s} ?(\mathrm{~b})\) What is the speed of the com then?

Short Answer

Expert verified
(a) The CoM is 22 m from the traffic light; (b) The speed of the CoM is 9.33 m/s.

Step by step solution

01

Find the Position of the Automobile at t = 3.0 s

The automobile starts from rest, so its initial velocity \( u = 0 \). Its acceleration \( a = 4.0 \mathrm{~m/s^2} \). We use the formula for motion under constant acceleration: \[ s = ut + \frac{1}{2}at^2 \] Substituting the values, \[ s_{\text{auto}} = 0 \times 3 + \frac{1}{2}(4.0) \times (3)^2 = 18 \mathrm{~m} \] The automobile covers 18 meters in 3 seconds.
02

Find the Position of the Truck at t = 3.0 s

The truck has a constant speed of \( 8.0 \mathrm{~m/s} \). Distance covered at constant speed is given by: \[ s = vt \] Substituting the values, \[ s_{\text{truck}} = 8.0 \times 3 = 24 \mathrm{~m} \] The truck covers 24 meters in 3 seconds.
03

Calculate the Center of Mass (CoM) Position

The center of mass for a system of two objects can be calculated as follows: \[ x_{\text{cm}} = \frac{m_1x_1 + m_2x_2}{m_1 + m_2} \] where \( m_1 = 1000 \text{ kg} \), \( x_1 = 18 \mathrm{~m} \), \( m_2 = 2000 \text{ kg} \), and \( x_2 = 24 \mathrm{~m} \). \[ x_{\text{cm}} = \frac{1000 \times 18 + 2000 \times 24}{1000 + 2000} = \frac{18000 + 48000}{3000} = 22 \mathrm{~m} \] Thus, the center of mass is 22 meters from the traffic light.
04

Find the Velocity of the Automobile at t = 3.0 s

The velocity of an object under constant acceleration can be found using: \[ v = u + at \] For the automobile, \[ v_{\text{auto}} = 0 + 4.0 \times 3 = 12 \mathrm{~m/s} \] The velocity of the automobile at 3 seconds is 12 m/s.
05

Velocity of the CoM

The velocity of the center of mass of a system is given by: \[ v_{\text{cm}} = \frac{m_1v_{1} + m_2v_{2}}{m_1 + m_2} \] where \( v_{1} = 12 \mathrm{~m/s} \) for the automobile and \( v_{2} = 8 \mathrm{~m/s} \) for the truck. \[ v_{\text{cm}} = \frac{1000 \times 12 + 2000 \times 8}{1000 + 2000} = \frac{12000 + 16000}{3000} = 9.33 \mathrm{~m/s} \] Hence, the speed of the center of mass is 9.33 m/s.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Motion Under Constant Acceleration
When a vehicle moves under constant acceleration, its velocity changes at a steady rate over time. This is a common scenario in physics, particularly when analyzing how objects behave when a force is applied. In our example, the automobile starts from rest, meaning its initial velocity
  • Initial velocity (\(u\)) is 0 m/s.
  • Acceleration (\(a\)) is 4.0 m/s虏.
To calculate how far it travels after a certain time (3 seconds in this case), we use the formula for distance (\(s\)) covered under constant acceleration:\[s = ut + \frac{1}{2}at^2\]Plugging in the values, we find the car travels 18 meters in 3 seconds. This clear calculation demonstrates how initial velocity and acceleration determine the future position of an object.It's essential to note that constant acceleration means the same amount is added to the velocity every second.
Center of Mass
The center of mass (CoM) is a critical concept that allows us to analyze the combined position of a system of particles or bodies in motion. It acts as the average position of all the mass in a system.For a car-truck system, the position of the CoM can be calculated using the formula:\[x_{\text{cm}} = \frac{m_1x_1 + m_2x_2}{m_1 + m_2}\]where
  • \(m_1\) and \(m_2\) are the masses of the car and truck respectively.
  • \(x_1\) and \(x_2\) are their respective positions.
By applying this formula, we found that the CoM is 22 meters from the traffic light after 3 seconds. This weighted average tells us where, on average, the system's mass is located in relation to a reference point.
Velocity Calculation
Velocity tells us how fast an object is moving and in what direction. When calculating the velocity of an object under constant acceleration, such as the automobile in our example, we use the formula:\[v = u + at\]The initial velocity (\(u\)) plus acceleration (\(a\)) times time (\(t\)) gives us the car's velocity after 3 seconds, which is 12 m/s.
  • Initial velocity was 0 m/s.
  • Acceleration was 4 m/s虏.
For the center of mass in a multi-object system, velocity can indicate how the entire system is moving. It is computed similarly to the position of CoM, using the weighted velocities of all components:\[v_{\text{cm}} = \frac{m_1v_1 + m_2v_2}{m_1 + m_2}\]This gives us insight into the overall speed and direction of the system.
Constant Speed
A constant speed implies that an object covers equal distances in equal intervals of time without changing its pace. For instance, the truck in this exercise moves with a constant speed of 8 m/s. This means it travels 8 meters every second.
  • No acceleration is acting on the truck.
  • Its speed remains unchanged because it's not speeding up or slowing down.
To find how far it travels in 3 seconds, we use:\[s = vt\]where \(v\) is the speed and \(t\) is the time. The truck covers 24 meters in this time frame. Understanding constant speed is fundamental for distinguishing it from scenarios involving acceleration, as it affects both the calculation of distance and time traveled.

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