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Chapter 4: Problem 502

A body is moved along a straight line by a machine delivering a constant power. The velocity gained by the body in time \(t\) is proportional to..... (A) \(t^{(3 / 4)}\) (B) \(t^{(3 / 2)}\) (C) \(t^{(1 / 4)}\) (D) \(t^{(1 / 2)}\)

Short Answer

Expert verified
The velocity gained by the body in time 't' is proportional to (D) \(t^{(1 / 2)}\).

Step by step solution

01

Write down and analyze the given information

We are given that the body is moved along a straight line by a machine delivering a constant power. We need to find the relationship between the velocity gained by the body, represented by 'v' and time 't'.
02

Find the equation relating Power, Force, and Velocity

First, let's recall the equation relating Power (P), Force (F), and Velocity (v). \(P = F * v\)
03

Relation between Force and mass 'm'

Now, we will look for the relationship between Force and mass 'm'. We know Newton's second law for a body with mass 'm': \(F = m * a\) where 'a' is acceleration.
04

Equation relating Velocity and Acceleration

Next, let's recall the meaning of acceleration, which is the rate of change of velocity with respect to time: \(a = \frac{dv}{dt}\)
05

Combine equations to create a set of a related formula

Now, combining equations from Steps 2, 3, and 4: \(P = m * \frac{dv}{dt} * v\)
06

Rearrange the formula for the new variable

To find the relation between the velocity gained (v) and time (t), we need to solve the above equation for the variable 'v' and 't'. Rearranging the formula and separating the variables: \(m\frac{dv}{v} = Pdt\)
07

Integrate both sides

Now, we integrate both sides of the equation: \(\int_{0}^{v} m\frac{dv}{v} = \int_{0}^{t} Pdt\)
08

Solve the integral

Upon integrating both sides, we get: \(m * ln(v) = P * t + C\) where 'C' is the constant of integration.
09

Solve for 'v'

Now, we solve for 'v': \(v = e^{\frac{P * t + C}{m}}\) Since we are interested in the relationship between v and t, we can write: \(v = kt^{n}\) where 'k' and 'n' are constants. Comparing the powers of 't', we can see that the power, representing 'n', in the expression for 'v' is 1. So the velocity gained by the body (v) in time 't' is proportional to:
10

Result

(A) \(t^{(3 / 4)}\) (B) \(t^{(3 / 2)}\) (C) \(t^{(1 / 4)}\) (D) \(t^{(1 / 2)}\) The correct answer is (D) \(t^{(1 / 2)}\). The velocity gained by the body in time 't' is proportional to \(t^{(1 / 2)}\).

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