Question

A rocket follows a path given by y equals x minus StartFraction 1 Over 65 EndFraction x cubed ?(distances in? miles). If the horizontal velocity is given by v Subscript x Baseline equals x comma find the magnitude and direction of the velocity when the rocket hits the ground? (assume level? terrain) if time is in minutes.

Answer 1

Sol::

Given
y = x - x³/65= x(1 - x²/65)

It touch the ground (y = 0) at 2 position corresponding to x = 0
And
x = √65 i.e x ≥ 0 (hit point)

Now let the velocity makes an angle θ with the x-axis,
tanθ is the slope of the tangent line ,

tanθ = dy/dx
= 1 - 3*(x²/65)
At the hit point, x = √65

tanθ = 1 - 3*(65/65)
= -2

Direction of v:

v makes with the x-axis an angle equal to

θ = tan^-1 (-2)
= -63.43°

Magnitude of v is equal to

v = v_x/cosθ

At hit point,
v_x = x = √65
So

v = √65/cos(-63.43°)
= 18.02 km/minute

Hope this helps you..
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