Velocity of object sliding down incline, given gravity and the incline's height

In summary, the problem is asking to determine the speed of a cart at the bottom of a hill with a height of 159m, given an initial speed of 0m/s and an acceleration of gravity of 9.8 m/s^2. Friction and air resistance are neglected. There is no mention of time. There are two possible approaches to solving this problem - using classical movement equations by assuming the angle of the incline and finding the distance and gravity force component parallel to the hill, or using energy conservation by assuming the mass of the object and equating the energy at the start and end of the movement.
  • #1
jstep
11
0

Homework Statement



a cart slides down an iced incline that is 159m high.

initial speed is 0m/s

acceleration of gravity is 9.8 m/s^2

neglect friction, air resistance

determine carts speed at the bottom of the hill

Homework Equations



v = u + at, but no time is mentioned

v^2 = u^2 + 2ah, if i treat it like freefall

The Attempt at a Solution



i don't hardly know where to start with this one, my first instinct was to break the two-dimensional motion into component vectors. but then i realized i don't have the length of the incline or even the angle of the incline, only the height.

i don't have any notion of time either, so i can't use v = u + at

i could treat it like freefall and use v^2 = u^2 + 2ah

v = sqrt(u^2 + 2ah), u = 0

v = sqrt((2)(9.8)(159))

v = 55.82 m/s

but i don't believe i can treat it like freefall, because aren't you supposed to be able to break up two-dimensional velocity into vectors that are completely independent of each other? and i believe the question is asking for the v in the x direction.

i'm not asking for an answer, i need to understand how to do it myself. but i don't think I'm approaching this problem with the right process. I think i just need a nudge in the right direction.

Thank you.
 
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  • #2
jstep said:

Homework Statement


v = u + at, but no time is mentioned
Thats why we love conservation laws. :smile:

What is constans during ride down?

regards
 
  • #3
the only thing that is constant is acceleration in the y-direction, acceleration in the x-direction is increasing. are you speaking of conservation of energy? I don't understand how that applies to this scenario... I don't even have the object's mass.
 
  • #4
jstep said:
the only thing that is constant is acceleration in the y-direction, acceleration in the x-direction is increasing. are you speaking of conservation of energy? I don't understand how that applies to this scenario... I don't even have the object's mass.
Energy is constans and of course I suggest use energy conservation law. To apply this you should assume, that you are know mass - it will be reduced.

The beauty of the conservation of energy law lies in the fact that no matter what happens in the meantime. No matter distance, time etc. You know energy at the start time, and at the end. They are equal. Thats all.

regards
Bartek
ps
of course you can solve this using classical movement equations. You have to assumed that you know the angle of hill, then find distance and qravity force component parallel to hill. No matter, that you in fact don't know that angle - it will be reduced.
 
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  • #5


Your approach of breaking the motion into components is correct. The key here is to recognize that the cart will only experience acceleration in the vertical direction due to gravity. Therefore, the initial velocity in the horizontal direction will remain unchanged throughout the motion.

To solve for the final velocity in the horizontal direction (v_x), you can use the equation: v_x = u_x + a_x * t, where u_x is the initial velocity in the horizontal direction (which is 0 m/s in this case), a_x is the acceleration in the horizontal direction (also 0 m/s^2 in this case), and t is the time taken for the cart to slide down the incline.

Since the problem does not provide any information about the time, you can use the equation v = u + at, where v is the final velocity in the vertical direction (v_y), u is the initial velocity in the vertical direction (also 0 m/s in this case), a is the acceleration due to gravity (9.8 m/s^2), and t is the time taken for the cart to slide down the incline.

Since we are only concerned with the final velocity in the horizontal direction (v_x), you can ignore the vertical velocity (v_y) in this case. Therefore, the equation becomes: v_x = u_x + a_x * t, which simplifies to: v_x = 0 + 0 * t, which means that the final velocity in the horizontal direction (v_x) is also 0 m/s.

In conclusion, the cart's speed at the bottom of the hill will be 0 m/s in the horizontal direction, and the speed in the vertical direction will be equal to the speed calculated using v^2 = u^2 + 2ah, which is 55.82 m/s.
 

1. What is the formula for calculating the velocity of an object sliding down an incline?

The formula for calculating the velocity of an object sliding down an incline is v = √(2ghsinθ), where v is the velocity, g is the acceleration due to gravity (9.8 m/s²), h is the height of the incline, and θ is the angle of the incline.

2. How does the angle of the incline affect the velocity of the sliding object?

The angle of the incline affects the velocity of the sliding object because it determines the component of gravity acting on the object. The steeper the incline, the greater the component of gravity pulling the object down, resulting in a faster velocity.

3. Can the velocity of an object sliding down an incline be greater than the initial velocity?

Yes, the velocity of an object sliding down an incline can be greater than the initial velocity if the incline is steep enough. This is because the object gains kinetic energy as it slides down the incline due to the work done by gravity.

4. How does the height of the incline affect the velocity of the object?

The height of the incline has a direct impact on the velocity of the object. The higher the incline, the greater the potential energy of the object. As the object slides down the incline, this potential energy is converted into kinetic energy, resulting in a higher velocity.

5. Is the velocity of an object sliding down an incline affected by the mass of the object?

According to the equation v = √(2ghsinθ), the velocity of an object sliding down an incline is not affected by the mass of the object. This means that objects with different masses will have the same velocity if they are sliding down the same incline with the same height and angle.

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