How high would a person need to jump from a 16 inch box to get over a 44 inch wall 7 fett away

Question

If choir risers are 16 inches high and the safety wall is 7 feet away and 44 inches high, how high would a person need to jump to go over the safety wall?

Answer 1

Perhaps, the trajectory from where the person jumps with initial height can be modeled as a parabolic function being the height as a function of the horizontal distance in which is:

 

h(x)=ax²+bx+c where a<0 due to the trajectory being an inverted parabola.

 

In order to come up with this function, you need to consider two sets of equation involving the initial velocity and the angle in which the person jumps.

 

v_x(t)=vo*cos(θ) which is the horizontal velocity. Hence, x(t)=v_o*t*cos(θ)

 

This makes t = x/v_o*cos(θ)

 

Next y(t)=-1/2*g*t²+v_o*t*sin(θ)+h_o

 

Next, substitute t in the last equation and doing some algebraic simplification in which is:

 

y=h(x)=-1/2*g*x²/(v_o*cos(θ))²+x*tan(θ)+h_o.

 

Now, let's go back to the original equation being h(x) = ax²-bx+c = -a(x-x_sw)² + h_sw  in which would be the equation of the trajectory the person would make in clearing the safety wall. x_sw = the horizontal distance of the safety wall in which the person must jump, and h_o = height of safety wall the person must clear.

 

Next, rewrite the equation h(x) = -ax² + 2ax_swx - ax_sw² + h_sw

This implies b = 2ax_sw

                  h_o =-ax_sw² +h_sw Hence, a = (h_sw - h_o)/x_sw²

 

Once you get what a is, then you could find b by substitute a in b = 2ax_sw = 2(h_sw - h_o)/x_sw.

 

Now with b known, one could find out the angle in which the person must jump by equating b to tan(θ) in which the angle is:

 

θ = tan-¹(b)

 

Therefore one could solve for the velocity in which the person must jump to clear the safety wall by equating 'a' to 1/2*g/(vo*cos(θ))². Hence, v_o = 1/cos(θ) * √(g/2a)

 

The question on whether a person could safely make the jump is dependent on whether the person could jump at the required initial velocity and angle to make the clearing.

 

I derived the formulas for you, and you could determine the required velocity and angle. I don't know what is the initial velocity in which an athlete could jump at.

 

I hope this helps.