Shooting two projectiles at the same time with different mass [closed]

Question

Shooting two projectiles having the same radius and in the same condition with the same initial velocity and angle one with a mass of 10kg and the second with a mass of 100kg which one of them will reach higher and further?

Answer 1

If you consider gravity alone, and neglect all other forces such as air resistance, the trajectories will be identical. But air resistance matters, at least a little and sometimes a lot. You can guess the answer by throwing a water balloon and a balloon full of air.

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There are two sources of air resistance. First flowing air generates friction forces. These are called viscous forces. Second, a moving object must push air in front of it out of the way. These are called inertial forces.

In most cases, one of these forces is so much bigger that you might as well ignore the other. There is a rule of thumb that tells you which is bigger.

Inertial forces are big when the projectile is big. Lots of air gets pushed around. Also when the projectile is fast, air must move fast to get out of the way. This takes a lot of energy and a lot of force to accelerate the air. And also when the fluid is dense. It takes more force and energy to move water out of the way than air.

Viscous forces are big when the fluid has lots of internal friction, or viscosity. You get more friction from push honey around than water.

Viscous forces are more important than inertial forces when a projectile is small. Loosely, this is because most of the friction takes place in a layer near the surface of the projectile. For a small projectile, the layer might be a big as the projectile. In a projectile with $10$ times the diameter, the layer has the same thickness. But the volume of water pushed out of the way is much larger.

The ratio of inertial to viscous forces is called the Reynolds number. While it is difficult to calculate exactly, it is easy to approximate. And an approximation is all you need when one force is overwhelmingly bigger than the other.

$$Re = \frac{\rho uL}{\mu}$$ where

• $\rho$ is the density of the fluid
• $u$ is the velocity
• $L$ is the size of the projectile
• $\mu$ is the viscosity of the fluid

For air and a projectile moving at everyday speeds, the rule of thumb is that inertial forces dominate for objects bigger than an insect. This is the case for most objects that people work with.

A projectile like a rock is usually bigger than an insect. So is a bullet, and it travels faster than everyday speeds. For a $10$ kg projectile, we don't even need to calculate. Inertial forces dominate.

It would be different if you wanted to know about which flea jumps farther.

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There is a very large force that you might not think about: Air pressure. And this is because when you are sitting still, forces from air pressure are equal on all sides and cancel. But things change when you are moving and pushing air out of the way, or flowing air generates friction. Air pressure is involved with both of these.

Air ahead of a projectile gets compressed as it is pushed. This increases air pressure. This pushes on the front of the projectile, which slows it. And it pushes air in front of the projectile, moving it out of the way.

Likewise behind the projectile would be vacuum, except that air pressure from air nearby pushes air in to fill the hole. Air is a little less dense behind the projectile. Air pressure from behind does push the projectile forward (A vacuum would not). But not as much as the air pressure in front pushes it backward. The net effect slows the projectile.

Since both projectiles have the same shape, these forces are the same for both.

Since $a = F/m$, these forces cause a larger acceleration on the lighter projectile. The lighter projectile is slowed more by air resistance, and reaches a lower altitude and hits the ground sooner.

Answer 2

When radius and all other conditions are the same, the greater mass will have greater kinetic energy which will take air resistance longer to slow it down. So if we take air resistance into account the 100kg projectile will go higher and further. If there were no air resistance so that g was the only acceleration acting on them after launch, then they would have the same maximum height and trajectory as different masses fall at the same rate neglecting air resistance.