Question #217755

It is required to project a body from a point on level ground in such a way

as to clear a thin vertical barrier of height h placed at a distance a from the

point of projection. Show that the body will just skim the top of the barrier

if

ga2

2u

2



tan2 α − a tan α +

ga2

2u

2

+ h



= 0

where u is the speed of projection and α is the angle of projection above the

horizontal


1
Expert's answer
2021-07-16T08:51:45-0400

Suppose that the motion starts from the origin and takes place in the .x; z/-plane,

where Oz points vertically upwards. Then the path of the body


z=xtanαg2u2cos2αx2z=x\tan \alpha-\frac{g}{2u^2\cos^2{\alpha}}x^2

where u is the projection speed and α\alpha is the angle between the direction of projection and the positive x-axis. If the path just skims the top of the

barrier, then u and α\alpha must satisfy the equation


h=atanαg2u2cos2αa2h=a\tan \alpha-\frac{g}{2u^2\cos^2{\alpha}}a^2

On using the trigonometric identity


sec2α=1+tan2α\sec^2\alpha=1+\tan^2\alpha

this condition can be written in the form


ga2tan2α2au2tanα+(ga2+2hu2)=0ga^2\tan^2\alpha-2au^2\tan \alpha+(ga^2+2hu^2)=0


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