Calculating the initial speed of an object that decelerates in a ramp.

[george ozua]

Homework Statement

Hello,
An object moves at constant speed in a horizontal surface. Suddenly, a ramp comes along its way. The object starts to climb such ramp. Due to this, the object starts to lose speed. At certain distance, the object loses all of its speed. I want to calculate the value of the object’s speed just before entering the ramp. Suppose that I know the value of the distance that the object travels on the ramp before stopping, the value of the coefficient of friction between the object and the ramp, the angle of elevation of the ramp and the value of acceleration due to gravity. I know that the following equation works for objects that decelerate in horizontal surfaces: velocity= the square root of: 2*distance*coefficient of friction*acceleration of gravity.
But, if the object decelerates in an elevated ramp, should I include trigonometry in the equation? I mean: should I multiply cos of the angle of elevation of the ramp, or sin or tan in the equation?

Homework Equations

velocity= the square root of: 2*distance*coefficient of friction*acceleration of gravity ¿(*tan, *sin, *cos)?

The Attempt at a Solution

It makes sense to me that trigonometric functions should be included inside the equation because the normal force of the object is altered due to the slope. But, if this is true, I have no idea which function to include and why. Thanks a lot!

 

[FactChecker]

You want a function that will agree with your horizontal equation when the angle of inclination is 0. You also want nearly zero deceleration due to friction when the elevated ramp is almost straight up. That is cos, since cos(0)=1 and cos(pi/2) = 0.

 

[NascentOxygen]

Homework Statement

Hello,
An object moves at constant speed in a horizontal surface. Suddenly, a ramp comes along its way. The object starts to climb such ramp. Due to this, the object starts to lose speed. At certain distance, the object loses all of its speed. I want to calculate the value of the object’s speed just before entering the ramp. Suppose that I know the value of the distance that the object travels on the ramp before stopping, the value of the coefficient of friction between the object and the ramp, the angle of elevation of the ramp and the value of acceleration due to gravity. I know that the following equation works for objects that decelerate in horizontal surfaces: velocity= the square root of: 2*distance*coefficient of friction*acceleration of gravity.
But, if the object decelerates in an elevated ramp, should I include trigonometry in the equation? I mean: should I multiply cos of the angle of elevation of the ramp, or sin or tan in the equation?

Hi george ozua. There are very few problems like this involving triangles which can be solved just by ruminating. The best (and therefore the recommended) approach is to first sketch a labelled diagram, this summarizes all the details known. Then mark on some forces, maybe resolve some into their components, and work towards a solution that way. It is sort of obvious that trig will be involved, but always stay alert to the possibility that some of it may cancel in the wash and leave your final answer even simpler than you'd maybe anticipated.

So, about that large clear diagram ...

 

[rude man]

Equate the initial kinetic energy to the sum of potential energy gained plus energy lost due to friction. You have all the parameters you need to solve this.

The elevation angle figures in both the the potential energy gain and the energy lost to friction. If you know the ramp distance, the ramp height should be obvious, yielding potential energy gain. And the frictional force is also dependent on that angle. Should it be sin, cos or tan? Well, pick the one that makes sense if the ramp angle were zero or 90 deg as well as your actual angle. Drawing a good diagram as nascent suggests is a very good idea.

 

[HalfLostAndSearching]

I would approach this problem by first understanding the physics involved. When an object is on a ramp, it experiences both a horizontal force (from its initial velocity) and a vertical force (from gravity). The angle of the ramp affects the balance of these forces, which in turn affects the object's motion.

To calculate the object's initial speed, we need to consider the forces acting on it and the laws governing those forces. In this case, we can use Newton's second law, which states that the net force on an object is equal to its mass times its acceleration (F=ma).

In this scenario, the object is initially moving at a constant speed, so its acceleration is zero. When it reaches the ramp, the force of gravity acting on the object will cause it to decelerate. The amount of deceleration depends on the angle of the ramp and the coefficient of friction between the object and the ramp.

To incorporate the angle of the ramp into our calculation, we can use trigonometry to break down the force of gravity into its horizontal and vertical components. The horizontal component will act to slow down the object, while the vertical component will not affect its motion along the ramp.

Therefore, the equation for calculating the initial speed of the object should include the cosine of the angle of the ramp, as it represents the horizontal component of the force of gravity. Your equation would then become:

velocity = √(2 * distance * coefficient of friction * acceleration of gravity * cosθ)

Where θ is the angle of the ramp.

I hope this helps you understand the physics behind the problem and how to incorporate trigonometry into the equation. Remember, as a scientist, it's important to always consider the underlying principles and laws governing a situation before applying equations or formulas.