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Chapter 4: Problem 7

In outer space, a constant force is applied to a \(32.5 \mathrm{~kg}\) probe initially at rest. The probe moves a distance of \(100 \mathrm{~m}\) in \(10 \mathrm{~s}\). (a) What acceleration does this force produce? (b) What is the magnitude of the force?

Short Answer

Expert verified
(a) The acceleration is \(2 \mathrm{~m/s^2}\). (b) The force is \(65 \mathrm{~N}\).

Step by step solution

01

Identify Given Values

First, let's list down what we know. The mass of the probe is \(m = 32.5 \mathrm{~kg}\), the distance it travels is \(d = 100 \mathrm{~m}\), and the time taken is \(t = 10 \mathrm{~s}\). The initial velocity of the probe is \(v_0 = 0 \mathrm{~m/s}\) since it starts from rest.
02

Use Kinematic Equation to find Acceleration

We will use the kinematic equation: \[ d = v_0t + \frac{1}{2}at^2 \]Since \(v_0 = 0\), the equation simplifies to \[ d = \frac{1}{2}at^2 \]Solving for \(a\), we get:\[ a = \frac{2d}{t^2} \]Substitute \(d = 100 \mathrm{~m}\) and \(t = 10 \mathrm{~s}\), \[ a = \frac{2 \times 100}{10^2} = \frac{200}{100} = 2 \mathrm{~m/s^2} \]
03

Apply Newton's Second Law to find the Force

Using Newton's second law, which states that force is equal to mass times acceleration (\( F = ma \)), we can calculate the force applied:\[ F = m \times a \]Substituting \(m = 32.5 \mathrm{~kg}\) and \(a = 2 \mathrm{~m/s^2}\),\[ F = 32.5 \times 2 = 65 \mathrm{~N} \].

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Key Concepts

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

Kinematic Equations
Understanding kinematic equations is fundamental in physics. These equations describe motion in terms of distance, velocity, time, and acceleration. They help us predict how an object moves under certain conditions. One of these equations is \[d = v_0 t + \frac{1}{2} a t^2\]This formula calculates the distance traveled (\(d\)) based on initial velocity (\(v_0\)), time (\(t\)), and acceleration (\(a\)). If an object starts from rest, as seen in our exercise, \(v_0\) becomes zero, simplifying the equation to:\[d = \frac{1}{2} a t^2\]By rearranging this expression, we can solve for acceleration. The simplified version helps us find the acceleration when the initial speed is zero by plugging in the values for distance and time. Recognizing how kinematic equations allow you to relate these variables is crucial for solving many physics problems.
Acceleration
Acceleration refers to the rate of change of velocity of an object. It tells us how quickly an object is speeding up or slowing down. In our exercise, the probe experiences acceleration due to a force acting on it in space. We calculate acceleration using the equation\[a = \frac{2d}{t^2}\]When we substitute the probe's distance (\(d = 100 \mathrm{~m}\)) and time (\(t = 10 \mathrm{~s}\)) into the equation, we determine the acceleration as beneficial to solve such motion problems. Hence, the probe's acceleration is:- \(2 \mathrm{~m/s^2}\).Understanding acceleration is essential for analyzing motion and predicting future behavior of moving objects.
Force Calculation
Force calculation is directly tied to Newton's Second Law. This law illustrates the relationship between force, mass, and acceleration. According to Newton's law, force (\(F\)) equals mass (\(m\)) times acceleration (\(a\)):\[F = ma\]In the exercise, we apply this principle to find the force acting on the probe. With a mass of \(32.5 \mathrm{~kg}\) and acceleration of \(2 \mathrm{~m/s^2}\), the force is calculated as:- \(65 \mathrm{~N}\).Understanding force calculation is vital in predicting how objects will move or react to forces applied or experienced. Newton's Second Law plays a huge role in numerous applications, from designing vehicles to launching spacecraft.

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Most popular questions from this chapter

A rifle shoots a \(4.20 \mathrm{~g}\) bullet out of its barrel. The bullet has a muzzle velocity of \(965 \mathrm{~m} / \mathrm{s}\) just as it leaves the barrel. Assuming a constant horizontal acceleration over a distance of \(45.0 \mathrm{~cm}\) starting from rest, with no friction between the bullet and the barrel, (a) what force does the rifle exert on the bullet while it is in the barrel? (b) Draw a free-body diagram of the bullet (i) while it is in the barrel and (ii) just after it has left the barrel. (c) How many \(g\) 's of acceleration does the rifle give this bullet? (d) For how long a time is the bullet in the barrel?

Human biomechanics. The fastest pitched baseball was measured at \(46 \mathrm{~m} / \mathrm{s}\). Typically, a baseball has a mass of \(145 \mathrm{~g}\). If the pitcher exerted his force (assumed to be horizontal and constant) over a distance of \(1.0 \mathrm{~m},\) (a) what force did he produce on the ball during this record-setting pitch? (b) Make free-body diagrams of the ball during the pitch and just after it has left the pitcher's hand.

Extraterrestrial physics. You have landed on an unknown planet, Newtonia, and want to know what objects will weigh there. You find that when a certain tool is pushed on a frictionless horizontal surface by a \(12.0 \mathrm{~N}\) force, it moves \(16.0 \mathrm{~m}\) in the first \(2.00 \mathrm{~s}\), starting from rest. You next observe that if you release this tool from rest at \(10.0 \mathrm{~m}\) above the ground, it takes 2.58 s to reach the ground. What does the tool weigh on Newtonia, and what would it weigh on Earth?

Two crates, \(A\) and \(B\), sit at rest side by side on a frictionless horizontal surface. The crates have masses \(m_{\mathrm{A}}\) and \(m_{\mathrm{B}}\). A horizontal force \(\vec{F}\) is applied to crate \(A,\) and the two crates move off to the right. (a) Draw clearly labeled free-body diagrams for crate \(A\) and for crate \(B\). Indicate which pairs of forces, if any, are third-law action-reaction pairs. (b) If the magnitude of force \(\overrightarrow{\boldsymbol{F}}\) is less than the total weight of the two crates, will it cause the crates to move? Explain.

A ball is hanging from a long string that is tied to the ceiling of a train car traveling eastward on horizontal tracks. An observer inside the train car sees the ball hang motionless. Draw a clearly labeled free-body diagram for the ball if (a) the train has a uniform velocity, and (b) the train is speeding up uniformly. Is the net force on the ball zero in either case? Explain.
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