/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 210 The velocity of a particle is \(... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 210

The velocity of a particle is \(v=v_{0}+g t+f t^{2}\). If its position is \(x=0\) at \(t=0\), then its displacement after unit time \((t=1)\) is (A) \(v_{0}+g / 2+f\) (B) \(v_{0}+2 g+3 f\) (C) \(v_{0}+g / 2+f / 3\) (D) \(v_{0}+g+f\)

Short Answer

Expert verified
(C) \(v_0 + \frac{1}{2}g + \frac{1}{3}f\)

Step by step solution

01

Since the velocity function is given by \(v= v_0 + gt + ft^2\), we can find the displacement function by integrating the velocity function with respect to \(t\). So, \(x(t) = \int (v_0 + gt + ft^2) dt\) #Step 2: Integrate the velocity function#

To find the displacement function, we integrate each term separately: \(x(t) = \int v_0 dt + \int gt dt + \int ft^2 dt\) \(x(t) = v_0t + \frac{1}{2}gt^2 + \frac{1}{3}ft^3 + C\) where C is the integration constant. #Step 3: Apply initial conditions#
02

We have the initial conditions \(x=0\) at \(t=0\). Let's use these conditions to find the value of the integration constant \(C\): \(x(0) = 0 = 0 + 0 + 0 + C\) So, \(C = 0\). Therefore, the displacement function becomes: \(x(t) = v_0t + \frac{1}{2}gt^2 + \frac{1}{3}ft^3\) #Step 4: Calculate displacement at t=1#

Now, we need to find the displacement at \(t = 1\): \(x(1) = v_0(1) + \frac{1}{2}g(1)^2 + \frac{1}{3}f(1)^3\) \(x(1) = v_0 + \frac{1}{2}g + \frac{1}{3}f\) Comparing with the given options, we find that the displacement at \(t = 1\) matches option (C): #Answer#: (C) \(v_0 + \frac{1}{2}g + \frac{1}{3}f\)

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Most popular questions from this chapter

The displacement of a particle is given by \(x=(t-2)^{2}\), where \(x\) is in metres and \(t\) in seconds. The distance covered by the particle in first \(4 \mathrm{~s}\) is (A) \(4 \mathrm{~m}\) (B) \(8 \mathrm{~m}\) (C) \(12 \mathrm{~m}\) (D) \(16 \mathrm{~m}\)

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