Velocity after Free Fall and after Motion down an Incline

In summary, there is a discrepancy between the manual and online sources regarding the speed of objects A and B at the bottom of their fall. The manual states that object A has a greater speed due to its formula v = sqrt(2gh), while the online sources argue that both objects have the same speed due to the conservation of energy. The discrepancy is attributed to a mistake in the manual's formula, which uses g(sinθ) instead of g as the acceleration on the incline.
  • #1
crastinus
78
9
I was doing some conceptual problems quickly before moving on to the numerical stuff when I glanced at this answer given in the solutions manual, and it surprised me:

Which object has greater speed at the bottom of its fall, object A of mass m dropped freely from height h or object B of mass m moving down an incline (no friction) from height h?

The manual says A. I googled around and found different people saying different things on this! I thought the answer was just that they have the same speed.

The manual gave this as its explanation: for A, v = sqrt(2gh), whereas for B, v = sqrt(2g(sinθ)h). Therefore, A is greater.

One of the online answers gave this: both A and B start with PE = mgh; so, at the end, all PE is now KE; so, KE = mgh = 1/2(mv^2); so both A and B have the same v, namely v = sqrt(2gh).

My analysis is that the manual mistakenly gives g(sinθ) for the acceleration on the incline. Maybe they were thinking of the force component?

I’d be grateful for help getting clear on this.
 
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  • #2
You are correct, and the manuel is incorrect.
 
  • #3
crastinus said:
v = sqrt(2g(sinθ)h)
That formula would be accurate for a ramp of [diagonal] length h, not for a ramp of [vertical] height h.
 
  • #4
crastinus said:
I thought the answer was just that they have the same speed.
Yes, as you can see from a simple application of the conservation of energy. In both cases the same amount of gravitational PE was lost, in both cases there were no friction losses, so all the PE goes into KE meaning they have the same KE and therefore the same speed.
 
  • #5
Thanks for the clarifications! I can see it more clearly now.
 

Related to Velocity after Free Fall and after Motion down an Incline

1. What is the formula for calculating velocity after free fall?

The formula for calculating velocity after free fall is v = gt, where v is the velocity in meters per second (m/s), g is the acceleration due to gravity (9.8 m/s²), and t is the time in seconds.

2. How does the velocity after free fall change with time?

The velocity after free fall increases as time increases. This is because the object is accelerating due to the force of gravity.

3. How does the angle of an incline affect the velocity after motion down the incline?

The steeper the incline, the faster the velocity after motion down the incline will be. This is because the force of gravity is acting more directly on the object, causing it to accelerate more quickly.

4. Is the velocity after motion down an incline affected by the mass of the object?

No, the velocity after motion down an incline is not affected by the mass of the object. This is because the force of gravity acts on all objects equally, regardless of their mass.

5. How does air resistance impact the velocity after free fall?

Air resistance can impact the velocity after free fall, as it is a force that opposes the motion of the falling object. However, its effect is usually negligible for objects with small mass or short distances of free fall.

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