# Calculating Mass for a Reduced Trajectory while Keeping Projectile Force Constant

Apologies if this is not a research-specific question, but applied mechanics.

A firefighter has to pass a test in which he is to throw a medicine ball a certain distance. The ball is thrown at a 45 degree angle (Fig. 1) and has to travel a ground distance of 7.5 meters. For the test, a medicine ball is a rough sphere with a mass of 4 kg and ~250 mm in diameter. In preparing for the test, however, the firefighter has only access to a dumbbell, which he does not intend to throw.

Given that a certain force is needed to throw the medicine ball with a preparation phase of 1 m and then released for a 7.5 meter throw, how heavy should the training substitute dumbbell be in order to achieve the equivalent force? Assume that the dumbbell only covers a 1 m distance at 45 degrees and that the firefighter intends to train at a 1 m/s velocity.

Figure 1

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For a projectile the range is given by $$R=\frac{u^2\sin2\theta}{g}$$ Where $u$ is the initial speed of projectile and $\theta$ is the angle of projection.
Substituting $R=7.5m, g=10 m/{s^2}, \theta = \frac \pi4$, we get $u$ as $$u=\sqrt{gR}=\sqrt{7.5 * 10} = 5\sqrt3 m/s$$ So the change in momentum of the ball is $$\Delta P = mu-0$$ $$\therefore \Delta P = 20\sqrt3 kgm/s$$ Change in momentum equals impulse applied by the thrower. Thus during training he must also practise to apply the same impulse. Change in momentum of the dumbbell will be $$\Delta P = m_{dumbbell}*1m/s$$ $$\therefore 1*m=20\sqrt3$$ $$m=20\sqrt3kg$$.