# Speed of a Body on a Smooth Surface over Time

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In summary, the problem involves a body of mass decreasing exponentially on a smooth surface. The mass is ejected backwards with a relative velocity and the question is asking for the speed of the body at a given time. This can be solved using the rocket equation, which involves conservation of momentum and an equation with logarithms and exponentials.
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Question:A body of mass is placed on a smooth horizontal surface. The mass of the body is decreased exponentially with disintegration constant λ. Assuming that the mass is ejected backwards with a relative velocity u.If initially the body
was at rest, the speed of the body at time t is:
(a)ue^(-t)
(b)uλt
(c)ue^(-λt)
(d)u{1-e^(-λt)}.

I guess this does not qualify as speedy, but do you know the rocket equation? What you have is a rocket. Figuring out the speed of a rocket is not trivial, but it all depends on conservation of momentum. A decent explanation is given here.

http://ed-thelen.org/rocket-eq.html

The final equation can be written as

$$v(t) = v_0 + uln \left[ \frac{M_0}{M(t)} \right]$$

In your problem, the initial velocity is zero. If you understand logs and exponentials, you can do the rest.

## What is the definition of speed?

Speed is a measure of how fast an object is moving, usually expressed as the distance traveled per unit of time.

## How is the speed of a body on a smooth surface calculated?

The speed of a body on a smooth surface can be calculated by dividing the distance traveled by the time it took to travel that distance.

## What factors can affect the speed of a body on a smooth surface?

The speed of a body on a smooth surface can be affected by factors such as friction, air resistance, and the slope of the surface.

## How does the speed of a body on a smooth surface change over time?

The speed of a body on a smooth surface can change over time depending on the forces acting upon it. If there is a net force acting on the body, its speed will change over time according to Newton's Second Law of Motion (F=ma).

## What is the relationship between speed and time for a body on a smooth surface?

The relationship between speed and time for a body on a smooth surface can be described by a linear equation, where the slope of the line represents the object's speed and the y-intercept represents its initial speed. This is known as the equation of motion for constant acceleration (v = v0 + at).

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