Skip to main content
Ch 06: Dynamics I: Motion Along a Line
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 06: Dynamics I: Motion Along a LineProblem 78c
Chapter 6, Problem 78c

A 1.0-cm-diameter, 2.0 g marble is shot horizontally into a tank of 20°C olive oil at 10 cm/s. How far in cm will it travel before stopping?

Verified step by step guidance
1
Determine the forces acting on the marble as it moves through the olive oil. The primary force opposing its motion is the drag force, which can be expressed as \( F_d = \frac{1}{2} C_d \rho A v^2 \), where \( C_d \) is the drag coefficient, \( \rho \) is the density of the fluid (olive oil), \( A \) is the cross-sectional area of the marble, and \( v \) is its velocity.
Calculate the cross-sectional area \( A \) of the marble using the formula for the area of a circle: \( A = \pi r^2 \), where \( r \) is the radius of the marble. The radius can be found by dividing the diameter (1.0 cm) by 2.
Write the equation of motion for the marble. The net force acting on the marble is \( F = ma \), where \( m \) is the mass of the marble and \( a \) is its acceleration. Since the drag force is the only force acting in the horizontal direction, \( ma = -F_d \). Substitute \( F_d \) into this equation to express the acceleration in terms of velocity: \( a = -\frac{1}{2m} C_d \rho A v^2 \).
Recognize that the velocity \( v \) decreases over time due to the drag force. Use the relationship between acceleration and velocity: \( a = \frac{dv}{dt} \). Substitute \( a \) from the previous step into this equation to obtain \( \frac{dv}{dt} = -\frac{1}{2m} C_d \rho A v^2 \). Rearrange to separate variables: \( \frac{dv}{v^2} = -\frac{1}{2m} C_d \rho A dt \).
Integrate both sides of the equation to find the distance traveled by the marble before it stops. The left-hand side integrates with respect to velocity from \( v = 10 \ \text{cm/s} \) to \( v = 0 \), and the right-hand side integrates with respect to time. After solving for time, use the relationship \( v = \frac{dx}{dt} \) to find the total distance traveled by the marble.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
7m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Drag Force

The drag force is the resistance experienced by an object moving through a fluid, such as olive oil. It depends on the object's velocity, the fluid's density, and the object's cross-sectional area. The drag force acts opposite to the direction of motion and increases with speed, ultimately affecting how far the marble will travel before stopping.
Recommended video:
Guided course
06:48
Intro to Centripetal Forces

Terminal Velocity

Terminal velocity is the constant speed an object reaches when the drag force equals the gravitational force acting on it, resulting in no net acceleration. In this scenario, the marble will decelerate due to drag until it reaches a point where the forces balance, which is crucial for determining how far it will travel in the olive oil.
Recommended video:
Guided course
7:27
Escape Velocity

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. They relate displacement, initial velocity, final velocity, acceleration, and time. In this problem, these equations can be used to calculate the distance the marble travels before coming to a stop, taking into account the deceleration caused by the drag force.
Recommended video:
Guided course
08:25
Kinematics Equations
Related Practice
Textbook Question

A 4.0-cm-diameter, 55 g ball is shot horizontally into a tank of 40°C honey. How long will it take for the horizontal speed to decrease to 10% of its initial value?

3
views
Textbook Question

An object with cross section A is shot horizontally across frictionless ice. Its initial velocity is v₀ₓ at t₀ = 0 s. Air resistance is not negligible. Show that the velocity at time t is given by the expression vx=v0x1+CdρAv0xt/2mv_x = \(\frac{v_{0x}\)}{1 + C_d \(\rho\) A v_{0x} t / 2m}.

2
views
Textbook Question

A block of mass m is at rest at the origin at t = 0. It is pushed with constant force F₀ from 𝓍 = 0 to 𝓍 = L across a horizontal surface whose coefficient of kinetic friction is μₖ = μ₀ ( 1 - 𝓍/L ) . That is, the coefficient of friction decreases from μ₀ at 𝓍 = 0 to zero at 𝓍 = L. b. Find an expression for the block's speed as it reaches position L.

Textbook Question

A spherical particle of mass m is shot horizontally with initial speed v₀ into a viscous fluid. Use Stokes' law to find an expression for vₓ (t), the horizontal velocity as a function of time. Vertical motion due to gravity can be ignored.

2
views