Calculating Force Required to Throw a 1kg Ball 4s in Air

[nessan]

Homework Statement

How much force is required to throw a 1kg ball (from the ground) so that it stays 4 seconds in the air before it touches the ground again. Air resistance is insignificant.

Given data:
m=1kg, t=4s, a=-9,8m/s² and ∆d=0 (because it hit the ground again)

Unknown data:
v_i (initial), h (the height of the ball when it turns down) and F (the required force)

Homework Equations

First equation $$\ mgh=\frac{mv_i^2}{2}$$
Second equation $$\Delta d=v_it+\frac{at^2}{2}$$
Third equation $$\ W=Fd$$

The Attempt at a Solution

By using the first equation i solved $$\ h=\frac{v_i^2}{2g}$$
and then I solved v_i using the second equation so $$\ v_i=\frac{\Delta d}{t}-\frac{at}{2}$$
since ∆d is 0, $$\ v_i=-\frac{at}{2}$$
to rewrite an equation for h without v_i, I wrote v_i as -at/2 so i got:
$$\ h=-\frac{at^2}{8}$$
With the third equation mgh=Fd I solved F:
$$\ F=-\frac{mgat^2}{d}$$
But d is the same as h so they cancel out and I get F=mg.

Can you please tell me what's wrong I'm doing and how to solve this problem.

Thanks in advance!

 

[gneill]

Your second equation describes the vertical displacement with respect to time given some initial velocity v_i. The displacement starts at zero, increases (upwards) until some maximum height is achieved, then falls back and eventually returns to Earth where the displacement is once again zero. If you plug in a zero for the displacement and set the time equal to your desired "hang time", you should be able to solve for the required initial velocity.

Now the problem becomes, how does the ball obtain this initial velocity? Apparently you want to apply some net force F to it, causing it to accelerate upwards. But acceleration occurs over some time and distance, neither of which has been specified in the problem statement. Put another way, the ball needs to be given an initial momentum m*v_i, which could be accomplished by applying a force F over time interval Δt. It would appear that there is not enough information given for you to calculate a specific value for the force.