Find Height of Ramp: H=1.0m, d=1.0m, θ=30°

[onelildustbunn]

Hi guys! This is due tomorrow and I'm finally breaking down to ask for help in figuring it out.

Homework Statement

"A small block is placed at height h on a frictionless ramp, which is inclined by an angle θ from the horizontal. Upon release, the block slides down the ramp then falls a distance H to the floor. A small hole is located a distance D from the end of the ramp. Assume the block is released from rest when it is in the position shown in the diagram."

I'm on part D:

Now let H=1.0 m, d=1.0 m, and θ=30°. Find the height h.


Here's the diagram given for the problem:

Here is from part C (which definitely leads into part D):

The teacher hinted that we should consider the box on the ramp and the box in the air as two semi-separate parts of the problem, in the style that you find one and use it to find the other.

Homework Equations

The kinematic equations:
V_f=V_i + aΔt
ΔS=V_iΔt + 0.5a(Δt)²
V_f²= V_i² +2aΔS

The Attempt at a Solution

What I think needs to happen: I need to find out everything I can about the freefall portion first, especially V₁ and V₂. Then because I know V₀ is 0 m/s (rest), and also that acceleration on the ramp is constant, I can find the distance traveled. Once I know the length of the ramp, simple trig can tell me its height.

My problems:
I haven't been given θ₁ or θ₂, which I need to calculate the velocities during freefall. How do I figure it out? Would I be wrong in thinking θ₁=θ₀, which I know is 30° (the ramp angle)?

Also, once I know everything about the freefall, how would I figure out the acceleration on the ramp? I don't know the object's mass so I can't use forces.

Any guidance appreciated! I'm not great with physics but I'm trying! )

 

[haruspex]

What I think needs to happen: I need to find out everything I can about the freefall portion first, especially V₁ and V₂.

The order doesn't matter. Each part will give you equations involving the speed of launch from the ramp (your v₁?).

.
I haven't been given θ₁ or θ₂,

Those being what angles?
From consideration of the forces while on the ramp, what is the acceleration down the ramp? Then, what are the horizontal and vertical components of it?

 

[onelildustbunn]

Thanks for responding! My comments about θ1 & θ2 probably didn't make sense because my second image didn't display with my post...sorry! Getting tired here (

I worked on the problem and came to an answer...this is what I did:

I found the height (h) of the ramp to be 0.25 m. Does this seem right? Did I commit any major no-no's? I felt uncertain about V_1y (velocity in the y direction) being 0 m/s, but I figured that if you think of the block at the instant it is off the ramp but before it begins to fall, it would only have velocity in the x direction, correct? It would have acceleration in the y direction of -g, but its initial velocity in the y direction would be 0. Was this an okay reasoning?

Thank you so much for any comments or guidance! )

 

[haruspex]

I found the height (h) of the ramp to be 0.25 m.

How, by solving the kinematic equations?

Does this seem right? Did I commit any major no-no's? I felt uncertain about V_1y (velocity in the y direction) being 0 m/s, but I figured that if you think of the block at the instant it is off the ramp but before it begins to fall, it would only have velocity in the x direction, correct?

No. It was moving down a ramp, and its velocity the instant after leaving the ramp will be the same as its velocity the instant before leaving.
Please, instead of making these wild leaps, try to follow the script. (There was another thread on the identical problem just recently, and I don't have another lifetime to spare.):
- Draw the free body diagram of the mass on the ramp
- Work out the acceleration in the down ramp direction
- Work out the horizontal and vertical components of that acceleration
- Use your kinematic equations and the unknown h to find the vertical velocity at end of ramp
- Then find the horizontal velocity at end of ramp
- From that, find the time to hit the hole
and we'll see where we go from there

 

[Nickie]

Hello,

To find the height h, we can use the kinematic equations and the information given in the problem.

First, let's consider the freefall portion of the block. We know that the block is released from rest, so the initial velocity (Vi) is 0 m/s. We also know that the block falls a distance H, so we can use the formula ΔS=ViΔt + 0.5a(Δt)2 to find the time it takes for the block to fall.

ΔS=H, Vi=0, a=g (acceleration due to gravity), so we have:

H=0(Δt) + 0.5(g)(Δt)2

Simplifying, we get:

Δt=√(2H/g)

Now, let's consider the ramp portion of the block's motion. We know that the ramp is inclined at an angle θ=30°, and the distance d=1.0 m. Using trigonometry, we can find the length of the ramp (L):

L=d/sinθ

L=1.0/sin(30°)

L=2.0 m

Now, we can use the formula Vf2=Vi2 +2aΔS to find the final velocity (Vf) of the block at the end of the ramp.

Vf2=0 +2(g)(L)

Vf=√(2gL)

Vf=√(2g)(2.0)

Vf=√(4g)

Vf=2√g

Now, we can use the formula Vf=Vi + aΔt to find the initial velocity (Vi) of the block at the top of the ramp.

Vf=2√g, Δt=√(2H/g), a=g

2√g=Vi + (g)(√(2H/g))

Solving for Vi, we get:

Vi=2√g - √(2gH)

Finally, we can use the formula Vf=Vi + aΔt to find the acceleration (a) on the ramp.

Vf=2√g, Vi=2√g - √(2gH), Δt=√(2H/g)

2√g=2√g