Accelerated Motion

Excerpts

• Accelerated Motion

  Thus, an impulse which changes the velocity from $50$ to $70$ feet per second, is twice as great as one which changes it from $50$ to $60$ feet.

• Accelerated Motion

  It is said to act uniformly, when the velocity acquired by the point in any one interval of time is the same as that acquired in any other interval of equal duration.

• Accelerated Motion

  Hence the limit to which we approximate by diminishing $t'$ without limit, is the length described in the time $t$ by a uniformly accelerated velocity, which shall increase from $0$ to $v$ in that time.

• Accelerated Motion

  *Force* is a name given to that which causes a change in the velocity of a body.

• Accelerated Motion

  It is plain that we cannot, by supposing any succession of impulses, however small, and however quickly repeated, arrive at a uniformly accelerated motion; because the length described between any two impulses will be uniformly described, which is inconsistent with the idea of continually accelerated velocity.

• Accelerated Motion

  In this substitute $v$ for $nv'$, and $t$ for $nt'$, which gives for the space described $\frac{1}{2}v(t + t')$. The smaller we suppose $t'$, the more nearly will this approach to $\frac{1}{2}vt$.

• Accelerated Motion

  And it must be observed that $6t$ is the differential coefficient of $3t^{2}$, or the coefficient of $dt$, in the development of $3(t + dt)^{2}$.

• Accelerated Motion

  But as the terms involving $(dt)^{2}$ in the velocities, etc., cannot be rejected without error, the above supposition of a uniform force cannot be made.

Equations

• Accelerated Motion

  nt' = t

  The time t is divided into n equal parts, each of length t', so n times t' equals t.

• Accelerated Motion

  nv' = v

  The velocity v is divided into n equal parts, each of size v', so n times v' equals v.

• Accelerated Motion

  n · \frac{(n + 1)}{2}\, v't' = \frac{n^{2} v't' + nv't'}{2}

  The sum 1 + 2 + ... + n equals n(n+1)/2, so the total space from n equal steps of v't' is (n^2 + n)/2 times v't'.

• Accelerated Motion

  \frac{1}{2}v(t + t')

  The space described in the stepwise motion, with n steps, is one half of v times (t + t').

• Accelerated Motion

  \frac{1}{2}vt

  As the interval t' is diminished without limit, the space described tends to one half of v times t, the length of uniformly accelerated motion from rest to velocity v in time t.

• Accelerated Motion

  v = gt

  With accelerating force g, the velocity acquired in t seconds from rest is g times t.

• Accelerated Motion

  \frac{1}{2}gt^{2}

  The space described from rest under uniform acceleration g in time t is one half of g times t squared.

• Accelerated Motion

  at + \frac{1}{2}gt^{2}

  With initial velocity a and uniform accelerating force g, the length described in time t is a·t plus one half of g·t².

• Accelerated Motion

  at - \frac{1}{2}gt^{2}

  With initial velocity a and uniform retarding force g, the length described in time t is a·t minus one half of g·t².

• Accelerated Motion

  a + gt

  The whole velocity after time t, with initial velocity a under uniform accelerating force g, is a plus g times t.

• Accelerated Motion

  \phi t = t^{3}

  The length described by the point in time t is t cubed, so the motion is defined by phi(t) equal to t cubed.

• Accelerated Motion

  3t^{2}

  The velocity of the point at the end of time t, for the motion phi(t) = t³, is 3t² inches per second, the coefficient of dt in the expansion of (t + dt)³.

• Accelerated Motion

  \phi' t\, dt + \phi'' t\, \frac{(dt)^{2}}{2} + \phi''' t\, \frac{(dt)^{3}}{2·3} + \etc.

  The length described in the interval dt equals phi(t + dt) minus phi(t), expanded as a series in dt with derivative coefficients.

• Accelerated Motion

  \phi' t\, dt + \frac{1}{2}\phi'' t (dt)^{2}

  The first two terms of the expansion represent the length described in dt with uniform velocity phi'(t) and accelerating force phi''(t), and approximate the motion for small dt as closely as we please.

Problems

No exercises in this chapter.