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Chapter 12: Problem 5

Given the acceleration of an object and its initial velocity, how do you find the velocity of the object, for \(t \geq 0 ?\)

Short Answer

Expert verified
Question: Given the acceleration function a(t) and the initial velocity V_0, find the velocity of the object for any time t 鈮� 0. Answer: To find the velocity of the object, integrate the acceleration function a(t) with respect to time and add the initial velocity V_0: \[V(t) = \int a(t) dt + V_0\]

Step by step solution

01

Understand the definitions

Acceleration is defined as the rate of change of velocity with respect to time, and it is mathematically represented as: \[\frac{dV(t)}{dt} = a(t)\] Where \(V(t)\) is the velocity of the object, \(a(t)\) is the acceleration and \(t\) is the time.
02

Integrate the acceleration function

We can find the velocity function by integrating the acceleration function with respect to time: \[V(t) = \int a(t) dt\] This will provide us with the velocity function \(V(t)\).
03

Add the initial velocity

After integrating the acceleration function, we'll get a constant term, \(C\), in our velocity function. To find the value of \(C\), we will use the initial velocity, given as \(V_0\). Since at \(t=0\), the velocity is \(V_0\), we can substitute this into our velocity function: \[V_0 = V(0) = \int_0^0 a(t) dt + C\] This implies that \(C=V_0\), so the final velocity function becomes: \[V(t) = \int a(t) dt + V_0\] Now, for any time \(t\geq 0\), just plug in the value of \(t\) into the velocity function, \(V(t)\), to find the velocity of the object.

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Most popular questions from this chapter

For the following vectors u and \(\mathbf{v}\) express u as the sum \(\mathbf{u}=\mathbf{p}+\mathbf{n},\) where \(\mathbf{p}\) is parallel to \(\mathbf{v}\) and \(\mathbf{n}\) is orthogonal to \(\mathbf{v}\). \(\mathbf{u}=\langle 4,3\rangle, \mathbf{v}=\langle 1,1\rangle\)

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