# Finding the Final Velocity

## Homework Statement

I'm going to be doing a lab, in which I am going to push a block down a ramp. So right now, the problem I'm going to ask is conceptual, and not actually with numbers.

So basically, I need to find the theoretical acceleration, and the theoretical velocity. Then, I need to find the actual velocity.

## Homework Equations

F = ma
Ff = μkFn
Fg = mg
Vf = Vi + aΔt
Δd = ViΔt + 1/2aΔt²

## The Attempt at a Solution

Alright, so here's my thoughts. I will be given the coefficient of kinetic friction. So, to find the acceleration, I'm going to try the following:

1. Find the distance of the ramp.
2. Time how long it takes to reach the bottom.
3. Weigh the block.
4. Plug in all the info I've found into the formula; Δd = ViΔt + 1/2aΔt²

So, that takes care of theoretical acceleration, I believe. If I take the theoretical acceleration, and plug it into the next formula; vf = vi + aΔt, then it will give me the theoretical final velocity.

So now the problem, finding the actual final velocity. I'm not exactly sure how I should do this one, but here's my thoughts;

If I measure out a certain length at the bottom of the ramp, I can time how long it takes for the block to travel that far. Then, using the average velocity formula, Δd = vΔt, I can find the number which is quite close to the actual velocity at the bottom of the ramp.

But is there a more accurate way of doing this? (And am I doing this right?)

There is also a much faster way to do this:

Measure (Theoretical value) the angle of the slope to the horizontal, mass of cart, the time it takes to travel the length of the ramp.

Calculate (Theoretical Value): The final velocity (v) of the cart, using suvat equations (u=0, time [measured], acceleration [calculated]). Acceleration is calculated from the Force parallel to the surface using the mass [measured] and angle of the ramp [measured].

Measure (Experimental value): Velocity at end of the ramp

Therefore the acceleration would be equal to gravity multiplied by sin(of the angle of the ramp)
and the final velocity would be calculated wither by
v=u+at
v^2=u^2+2as

where t is time and s is displacement and a is the acceleration which was previously found.

Also in orer to find the final velocity of the plack you can take a video of it going down the ramp and then analyze it through any of the video analysis programs in order to find the final velocity (tracker) is a free and easy to use option.

In the absence of friction the acceleration will be g sin θ, however if this is a block and not a cart with only rolling friction, then the acceleration of the block will be g (sin θ - μ cos θ) provided of course that tan θ > μ. Thirty years ago we used air tracks, gliders, and timers but now we use carts, tracks, and ultrasonic motion detectors tied to computers and plot out the position, velocity, and acceleration vs time. Much easier to analyze.

If you have studied the conservation of energy, this is a good application. The initial potential energy, mgh, equals the final kinetic energy, mv^2/2 plus the work done against friction, μ mg cos θ x.

haruspex
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1. Find the distance of the ramp.
2. Time how long it takes to reach the bottom.
3. Weigh the block.
4. Plug in all the info I've found into the formula; Δd = ViΔt + 1/2aΔt²

So, that takes care of theoretical acceleration,
No, if you are meauring those things and using the equation to find a then that is finding the actual acceleration.
How do you propose to find the theoretical acceleration? You wrote 'push' a block down a ramp. How do you know what force you are applying?

CWatters