# Distance Covered by Particle from A to +1 on Curve y=ax2

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In summary, a particle is released from a curve and slides down and flies out on a smooth surface. The equations of motion and conservation of energy are used to find the total horizontal distance traveled by the particle. The mistake in the solution is not using the projectile motion equations and trigonometry to find the components of velocity.
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## Homework Statement

A particle is released from a curve y=ax2(a=1m-1) from A on smooth surface. It slides down and flies out from +1 on +tve x . find total horizontal distance D traveled by it .
http://img59.imageshack.us/img59/6295/sumaj.png

## Homework Equations

Equations of motion
LCE

## The Attempt at a Solution

y1 = 1 * -22 = 4
y2 = 1 * 1 = 1
I used conservation of energy here.
mg * 4 = m*g*1 + 1/2 m * u2

Then got u = (60)1/2

Again applying LCE i get
mg * 4 = 1/2 * m * v2

From this v = (80)1/2

Now using v2 = u2 + 2*g*s

80 = 60 + 20s
s=1 metre

= 4 = D

But that's not the actual answer. Any suggestions and can you please point out my mistake?

Last edited by a moderator:
The first part to calculate the velocity at x=1m is correct. From that point you need to use your equations of motion and treat it like a projectile problem.

what about the components of velocity ... any hint on how to find them?

If this is a calc based class just take the derivative of ax^2 and evaluate it at x=1m, that will give you the slope at that point. Since slope is equal to y/x what trig function is equal to that as well?

oh right ... thanks!

## 1. What does the term "distance covered" mean in this context?

The term "distance covered" refers to the total length of the curve that the particle travels along from point A to point +1. It is a measure of the total displacement of the particle along the curve.

## 2. How is the distance covered by the particle calculated?

The distance covered by the particle can be calculated using the arc length formula for a curve, which takes into account the changing slope of the curve at each point. It involves integrating the square root of the sum of the squares of the first and second derivatives of the curve with respect to the variable x.

## 3. Can the distance covered by the particle be negative?

No, the distance covered by the particle cannot be negative. It is always a positive value, as it represents the total length of the curve traveled by the particle.

## 4. How does the value of the coefficient a affect the distance covered by the particle?

The value of the coefficient a affects the shape of the curve y=ax^2, which in turn affects the distance covered by the particle. A larger value of a leads to a steeper and narrower curve, resulting in a longer distance covered by the particle. Conversely, a smaller value of a results in a flatter and wider curve, leading to a shorter distance covered by the particle.

## 5. Does the distance covered by the particle change if the starting point at A is different?

Yes, the distance covered by the particle will change if the starting point at A is different. This is because the starting point affects the total length of the curve that the particle will travel along to reach point +1. A different starting point will result in a different total displacement and therefore a different distance covered by the particle.

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