A Point Charge in an Electric Field — Question 8

Question 8

A particle can strike a plate even when its predicted exit height is zero

A particle enters midway between parallel boundaries at $y=\pm h$ with upward velocity $u>0$. Its constant vertical acceleration is $-a$ with $a>0$. The horizontal flight time through the field region is T, chosen so the unobstructed trajectory would have $y(T)=0$.

1. Find a in terms of u,T, the time of maximum height and that height.

2. For $u=10.0$ m/s, $T=2.00$ ms and $h=4.00$ mm, determine whether the particle clears the region. Find the first impact time if not.

Question 8 - Solution

A particle can strike a plate even when its predicted exit height is zero

(a) Method and calculation

$$y(t)=ut-\tfrac12at^2,\quad y(T)=0\Rightarrow a=2u/T.$$$$t_{\max}=u/a=T/2,\quad y_{\max}=uT/4=5.00\,\mathrm{mm}>h.$$

(b) Interpretation and further calculation

$$a=1.00\times10^4\,\mathrm{m/s^2},\quad ut-\tfrac12at^2=h.$$$$t_{\mathrm{hit}}=\frac{u-\sqrt{u^2-2ah}}a\approx.552786\,\mathrm{ms}.$$The particle hits the upper boundary before the predicted field-free exit; the later ideal trajectory is no longer physically realized.

Physical check

Clearance requires checking the entire path, including interior extrema, rather than only its endpoints.