A particle P in an inertial reference frame has an initial velocity vo at the place xo, and subsequently moves under the influence of a force that is proportional to the time and acts in a fixed direction e. Find the position and velocity of P at time t .

Question

A particle P in an inertial reference frame has an initial velocity [latex]\mathbf{v}_0[/latex] at the place [latex]\mathbf{x}_0[/latex], and subsequently moves under the influence of a force that is proportional to the time and acts in a fixed direction e. Find the position and velocity of P at time t .

Accepted Answer

The force on P is given by [latex]\mathbf{F}(P, t)=k t \mathbf{e}[/latex] , where k is a constant and e is a constant unit vector. Use of this relation in (6.1) and integration of the result as shown in (6.20) with the initial value [latex]\mathbf{v}(P, 0)=\mathbf{v}_0[/latex] yields the velocity [latex]\mathbf{v}(P, t)=k t^2 / 2 m \mathbf{e}+\mathbf{v}_0 [/latex]. With the initial value [latex]\mathbf{x}(P, 0)=\mathbf{x}_0[/latex], a second integration described by (6.21) yields the motion [latex]\mathbf{x}(P, t)=k t^3 / 6 m \mathbf{e} + \mathbf{v}_0 t + \mathbf{x}_0[/latex]. Let the reader show that if P starts at the origin with velocity [latex]\mathbf{v}_0=v_0 \mathbf{j}[/latex] and the force acts in the direction [latex]e=i[/latex], the path of P is a cubic parabola [latex]x=c y^3[/latex]. Identify the constant c.

[latex]\mathbf{F}=\dot{\mathbf{p}}=m \mathbf{a}=m \dot{\mathbf{v}}=m\ddot{\mathbf{x}}[/latex] (6.1)

[latex]\mathbf{v}(P, t)=\frac{1}{m} \int \mathbf{F}(P, t) d t + \mathbf{c}_1[/latex] (6.20)

[latex]\mathbf{x}(P, t)=\int \mathbf{v}(P, t) d t + \mathbf{c}_2[/latex] (6.21)

Question 6.7: A particle P in an inertial reference frame has an initial v...

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