Wrecking ball

Relevant wiki: Conservation of Energy

Say the wrecking ball, which has mass $m,$ is released from a height of $h_1$ with respect to some arbitrary reference line, and at the point of impact it is at a height of $h_2 < h_1$ (under the same reference line) and has a velocity of $v.$ Since there is no air resistance, the potential energy of the ball at the start is equal to the sum of the potential and kinetic energies of the ball at the end, by the law of conservation of energy. Thus, we can write

$\begin{aligned} U_0 &= U_f + K_f \\ mgh_1 &= mgh_2 + \dfrac{1}{2}mv^2 \\ mg(h_1 - h_2) &= \dfrac{1}{2}mv^2 \\ v &= \sqrt{2g(h_1 - h_2)}. \end{aligned}$

Notice that the final velocity of the ball does not depend on its mass. Therefore, as long as we keep the initial conditions the same, the ball will hit at the same speed regardless of weight.

Relevant wiki: Newton's Law of Gravity - Problem Solving

The famous Galileo's Leaning Tower of Pisa experience (though it might not actually have been carried out) clearly illustrated that acceleration due to gravity is independent of mass. Therefore, neglecting air resistance, wrecking balls of different masses going through the same trajectory have the same velocity throughout the trajectory including the lowest point.

The solutions given cover the problem very nicely. I would just like to add two remarks:

• The main reason that the parameter of mass does not change the motion of the ball is that the potential energy is porpotional to the mass so we can write the energy as $E=m\left(\dfrac{u^2}{2}+gh\right)$ . The same would happen for example at an oscillator where you have potential energy $\sim m$ and specificaly $m\omega ^2 x^2 /2$ .

• This problem is a very nice example of the difference between the velocity and the momentum. Someone could ask himself if the velocity is the same then why do we use massive wrecking balls to destroy buildings? And from experience we know we would not even think to start wrecking a building with a basket ball. The difference is that we need big momentum $p = m u$ of the ball that hits the building. That momentum is transfered to the building during the coilision. Then from newton's second law $F=\dfrac{dp}{dt}$ , we make a big change in momentum of the walls of the building in a very short time so that is the reason a great force is created and we wreck the buildings.

Suppose your object is a sphere with a radius r and mass m. The aerodynamic drag on a sphere is given by:

$F_{drag} = \frac{1}{2}Cd \rho \pi r^2 v^2$ ...................................(1)

where $\rho$ is the density of the air and Cd is the drag coefficient. The drag coefficient varies with speed but over a limited range of speeds it can usefully be taken as constant.

So, the downward force on the object is simply:

$F_{grav} = mg$ ................................................(2)

and terminal velocity is reached when the two forces are in balance i.e. when $F_{drag} = F_{grav}$ . If we equate equations (1) and (2) we get:

$\frac{1}{2}Cd \rho \pi r^2v^2 = mg$

and rearranging them gives:

$V_{term} = \sqrt\frac{2mg}{Cd \rho \pi r^2}$

In your case you keep the size of the spheres constant, in which case we get:

$V_{term} \propto m$

So terminal velocity does increase with mass. The heavier sphere will have a higher terminal velocity. but in the case of law of conversion energy, we know that, $V=\sqrt{2gh}$ so, for this case mass does not create any change in the velocity.

It's pretty simple. Air ressistance is not effected by the mass of the object moving through it. An empty egg will fall at the same speed as a full one.

Since v(max) is independent of mass M, as v(max) = √(2gl(1-cos(theta))) where, theta = angle of release of the ball, l = length of rope, g = acceleration due to gravity.

our intuition tells us that the wrecking ball hits harder, and it does; but that is only because it is heavier. momentum is mass x velocity. heavier objects have more momentum, which means, it takes more to stop them. they hurt more when they hit you. the maximum speed is determined by the acceleration, which comes from gravity. we all know the classic physics demo; a hammer or a feather. in a perfect vacuum (no air resistance) both fall at the same rate.

A classic lab in high school physics is to put a steel ball bearing in a clear cylinder with a feather, remove all air (create a vacuum to eliminate air resistance) and drop both items. The observation is that both objects land at the same time proving mass has zero effect on gravitational acceleration.

Because Galileo.

I've been prompted for a more complete explanation. One word is sufficient.