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

Knight Calc 5th Edition

Ch 06: Dynamics I: Motion Along a Line

Problem 60

Chapter 6, Problem 60

A particle of mass m moving along the x-axis experiences the net force Fₓ = ct, where c is a constant. The particle has velocity v₀ₓ at t = 0. Find an algebraic expression for the particle's velocity vₓ at a later time t.

Verified step by step guidance

Start by recalling Newton's second law of motion, which states that the net force acting on an object is equal to the rate of change of its momentum. For a particle of constant mass m, this simplifies to:

F_x = m \(\frac{dv_x}{dt}\)

Substitute the given force

F_x = ct

into the equation:

m \(\frac{dv_x}{dt}\) = ct

. Rearrange to isolate

\(\frac{dv_x}{dt}\)

\(\frac{dv_x}{dt}\) = \(\frac{ct}{m}\)

Integrate both sides with respect to time to find the velocity

v_x

. The left-hand side becomes

\(\int\) dv_x

, and the right-hand side becomes

\(\int\) \(\frac{ct}{m}\) dt

v_x = \(\int\) \(\frac{ct}{m}\) dt

Perform the integration on the right-hand side. Since

c

and

m

are constants, they can be factored out:

v_x = \(\frac{c}{m}\) \(\int\) t dt

. The integral of

t

\(\frac{t^2}{2}\)

, so:

v_x = \(\frac{c}{m}\) \(\cdot\) \(\frac{t^2}{2}\) + C

, where

C

is the constant of integration.

Determine the constant of integration

C

using the initial condition. At

t = 0

, the velocity is

v_{0x}

. Substituting these values into the equation:

v_{0x} = \(\frac{c}{m}\) \(\cdot\) \(\frac{(0)^2}{2}\) + C

, we find

C = v_{0x}

. Thus, the final expression for the velocity is:

v_x = \(\frac{c}{2m}\) t^2 + v_{0x}

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Newton's Second Law of Motion

Newton's Second 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 expressed as F = ma, where F is the net force, m is the mass, and a is the acceleration. Understanding this law is crucial for analyzing how forces affect the motion of a particle.

Kinematics and Velocity

Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause the motion. Velocity is defined as the rate of change of displacement with respect to time. In this context, knowing how to relate acceleration to changes in velocity over time is essential for deriving the particle's velocity from the given force.

Integration in Physics

Integration is a mathematical process used to find the total accumulation of a quantity, such as velocity from acceleration. In physics, when a force leads to acceleration, integrating the acceleration function over time provides the change in velocity. This concept is vital for solving the problem, as it allows us to derive the velocity expression from the given force function.

Recommended video:

Guided course

11:43

Finding Moment Of Inertia By Integrating

Related Practice

Textbook Question

The 2.0 kg wood box in FIGURE P6.58 slides down a vertical wood wall while you push on it at a 45° angle. What magnitude of force should you apply to cause the box to slide down at a constant speed?

views

Textbook Question

Astronauts in space 'weigh' themselves by oscillating on a spring. Suppose the position of an oscillating 75 kg astronaut is given by $x = (0.30 m) \sin ((π rad/s) \times t)$ , where t is in s. What force does the spring exert on the astronaut at (a) t = 1.0 s and (b) 1.5 s? Note that the angle of the sine function is in radians.

views

Textbook Question

A 1.0 kg wood block is pressed against a vertical wood wall by the 12 N force shown in FIGURE P6.57. If the block is initially at rest, will it move upward, move downward, or stay at rest?

Textbook Question

A 500 g ball moves horizontally with velocity v_𝓍 = ( 15 m) / (t + 1 s) for t > 0 s. What is the net force on the ball at t = 1 s?

Textbook Question

A ball is shot from a compressed-air gun at twice its terminal speed. What is the ball's initial acceleration, as a multiple of g, if it is shot straight up?

views

Textbook Question

What is the magnitude of the acceleration of a skydiver at the instant she is falling at one-half her terminal speed?

views