Q about final velocity after dropping from same height

[bluecabinet]

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if I were to

1. drop a ball "m" in a free fall
2. drop a ball on a frictionless inclined plane of angle 30
3. drop a ball on a frictionless inclined plane of angle 60

would all of them have the same final velocity given that they were all drop from the SAME height?

 

[GomerPyle]

if I were to

1. drop a ball "m" in a free fall
2. drop a ball on a frictionless inclined plane of angle 30
3. drop a ball on a frictionless inclined plane of angle 60

would all of them have the same final velocity given that they were all drop from the SAME height?

I don't think they would because the downward acceleration of the ball in free fall is obviously just 'g', but the downward acceleration of the ball on an inclined plane is g*sin(theta) which is less than the acceleration of the ball in free fall. Thus it would take longer for the balls rolling down an inclined plane to hit the ground from the same height, and since v = change in x/change in time (change in x is constant in all 3 cases since they are dropped from the same height), the final velocity would be greater when the ball is in free fall rather than it sliding down a plane.

Remember that downward motion/velocity determines the time it takes for an object to hit the ground, and the horizontal motion components determine the distance the object travels. Thus the horizontal components of the inclined plane cases do not have any effect on the time it takes for the balls to reach the ground, only the vertical component does (which is gsin(theta)).

I am pretty sure I am correct, but if I am not, somebody correct me!

 

[Ineedhopenow]

Yes they would. Even though the forces on the inclined planes differ, the distance traveled also differs. 30 degree plane experiences the least force but travels a linger distance. The final velocity for all of them is the square root of 2gh.

 

[GomerPyle]

Yes they would. Even though the forces on the inclined planes differ, the distance traveled also differs. 30 degree plane experiences the least force but travels a linger distance. The final velocity for all of them is the square root of 2gh.

The vertical distance however doesn't change. The vertical component has less gravitational acceleration, which means it would have less velocity when it hits the ground relative to the ball in free fall. The horizontal component shouldn't have any effect on the final velocity with which the ball hits the ground.

 

[Ineedhopenow]

The vertical distance however doesn't change. The vertical component has less gravitational acceleration, which means it would have less velocity when it hits the ground relative to the ball in free fall. The horizontal component shouldn't have any effect on the final velocity with which the ball hits the ground.

You're right, the vertical distance doesn't change, but the distance the object travels does. The object on the ramp doesn't experience any horizontal acceleration, but it does experience a "fraction" of the vertical force. This force "fraction" over a longer distance (ramp distance > vertical distance) balances the velocities, because although the object experiences a smaller force, it travels a greater distance.

 

[GomerPyle]

You're right, the vertical distance doesn't change, but the distance the object travels does. The object on the ramp doesn't experience any horizontal acceleration, but it does experience a "fraction" of the vertical force. This force "fraction" over a longer distance (ramp distance > vertical distance) balances the velocities, because although the object experiences a smaller force, it travels a greater distance.

I still don't understand your reasoning.

If you drop a rock from a height of 10 meters and you threw a rock horizontally at the same time from the same height (10 meters), both rocks will land at the same time because the vertical component of motion is the same in both cases (- 9.8 m/s). The rock being thrown horizontally (and thus traveling a greater net distance) had no effect on the timing that the rock lands on the ground. Thus both rocks had to have landed with the same vertical velocity.

However in this case, a ball rolling down a plane has less vertical acceleration downwards (gsin(theta)), and using the same principles as stated above, the the net distance the ball travels doesn't affect the time it takes for the ball to reach the ground. Only the vertical component of motion does - and since the vertical acceleration is less for the balls traveling down a plane, these balls will land later than the ball dropped in free fall. So the time it takes to reach the ground is greater when the ball is on the plane, correct?

So I assume your saying that the longer time makes up for the smaller vertical acceleration, correct? Meaning that this fraction of longer time/ smaller acceleration is equal to the shorter time/ larger acceleration of the ball in free fall? Is this the assumption your making?

 

[dmf2682]

Here's what you do.

Consider it in terms of energy.

You start at rest from the same height in all three situations. Same mass, and same planet.

Pe = mgh

At the bottom of the fall, consider energy in the three situations.

A) fell straight down. All potential energy has been converted to kinetic energy.

B) fell at an angle. There's no losses in energy due to friction, but if there were, the energy lost would be (friction force)*d where d is the length of the path. In our case though all the pe is converted to ke.

C) same as b.

So because ke is the same in all three situations velocity is the same.

If you take the example someone else asked about, where a rock is falling versus a rock being thrown, the rock being thrown will have a starting ke and pe and will thus have a larger ke and final velocity than the one released from rest.

 

[Ineedhopenow]

Here's what you do.

Consider it in terms of energy.

You start at rest from the same height in all three situations. Same mass, and same planet.

Pe = mgh

At the bottom of the fall, consider energy in the three situations.

A) fell straight down. All potential energy has been converted to kinetic energy.

B) fell at an angle. There's no losses in energy due to friction, but if there were, the energy lost would be (friction force)*d where d is the length of the path. In our case though all the pe is converted to ke.

C) same as b.

So because ke is the same in all three situations velocity is the same.

If you take the example someone else asked about, where a rock is falling versus a rock being thrown, the rock being thrown will have a starting ke and pe and will thus have a larger ke and final velocity than the one released from rest.

Bravo! Conservation of energy is a great way to think of this problem.

 

[milski]

Same speed or magnitude of velocity. The velocity vectors will be pointed in different directions. Happy nit-picky morning. 😉

Treating it as an energy problem is the simplest way although doing the math for the acceleration components will work too.

 

[GomerPyle]

If you take the example someone else asked about, where a rock is falling versus a rock being thrown, the rock being thrown will have a starting ke and pe and will thus have a larger ke and final velocity than the one released from rest.

So even though the rocks hit the ground at the same time, the final velocity of the rock being thrown horizontally is greater than the rock that was just dropped?

 

[dmf2682]

So even though the rocks hit the ground at the same time, the final velocity of the rock being thrown horizontally is greater than the rock that was just dropped?

Before they hit the ground, yes. After they hit the ground though, both have zero velocity 🙂