Find the response (in terms of its acceleration, velocity, and position versus time) of a 3000-lb automobile to a force Fi of 500 lb, which is suddenly applied by three members of the football squad (i.e., a step change in force occurring at t = 0), ignoring friction effects (see Fig. 2.3). Assume

Question

Find the response (in terms of its acceleration, velocity, and position versus time) of a 3000-lb automobile to a force [latex]F_i[/latex] of 500 lb, which is suddenly applied by three members of the football squad (i.e., a step change in force occurring at t = 0), ignoring friction effects (see Fig. 2.3). Assume that the football players are able to maintain the applied force of 500 lb regardless of how fast the automobile moves. Reference [latex]r_1[/latex] is the local ground reference – in other words, the starting point for vehicle motion.

Accepted Answer

Using Eq. (2.1) –that is, ignoring friction effects –

[latex]m\left(\frac{dv_1}{dt} \right)=F_m [/latex], (2.1)

we have

[latex]\frac{dv_1}{dt}=\left(\frac{1}{m} \right) F_i(t)[/latex]. (2.3)

Because the vehicle acceleration [latex]a_1(t) = dv_1/dt,[/latex] we see that it undergoes a step change from 0 to (32.2)(500)/3000 at t = 0 and remains at that value until the applied force is removed. Next, we may separate variables in Eq. (2.3) and integrate with respect to time to solve for [latex]v_1= (t),[/latex]

[latex]\int_{v_1(0)}^{v_1(t)}{dv_1}=\left(\frac{1}{m} \right) \int_{0}^{t}{F_i}(t)dt [/latex], (2.4)

which yields

[latex]v_1(t)-v_1(0)=\frac{(32.2)}{6} t-0[/latex]

or

[latex]v_1(t) = v_1(0) + 5.37t[/latex] ft s; i.e., a ramp starting at t = 0 having a slope of [latex]5.37 ft/s^2.[/latex] Similarly, a second integration (assuming the initial velocity is zero) with respect to time yields

[latex]\int_{x_1(0)}^{x_1(t)}{dx}=5.37\int_{0}^{t}{tdt} [/latex] (2.5)

or
[latex]x_1(t) = x_1(0) + 2.68t^2 ft; i.e., a \ parabola \ starting \ at \ t = 0[/latex] . (2.6)

The results are shown as functions of time in Fig. 2.4 along with the input force [latex]F_i.[/latex]
Note that it takes time to build up changes in velocity and displacement because of the integrations involved. Example 2.1 displays one of the first indications of the role played by integration in determining the dynamic response of a system. From another point of view, the action of the applied force represents work being done on the mass as it accelerates, increasing the kinetic energy stored in it as time goes by. The rate at which energy is stored in the system is equal to the rate at which work is expended on it by the members of the football squad (the first law of thermodynamics):

[latex]\frac{dE _K}{dt} =F_iv_1[/latex] . (2.7)

Separating variables and integrating with respect to time, we find

[latex]\int_{E _k(0)}^{E _k(t)}{dE_K}=F_iv_1dt=\int_{0}^{t}{mv_1}\left(\frac{dv_1}{dt} \right)dt=m\int_{v_1(0)}^{v_1(t)}{v_1(t)dv_1}, [/latex]

so that

[latex]E _K(t)=E _K(0)+\left(\frac{m}{2} \right)v_1(t)^2. [/latex] (2.8)

Thus the stored energy accumulates over time, proportional to the square of the velocity, as the work is being done on the system; and the mass is an A-type element storing energy that is a function of the square of its A variable [latex]v_1[/latex] .

As this text proceeds to the analysis of more complex systems, the central role played by integration in shaping dynamic system response will become more and more evident.
It should be noted in passing that it would not be reasonable to try to impose on the automobile a step change in velocity, because this would be an impossible feat for three football players – or even for 10 million football players! Such a feat would require a very, very great (infinite) force as well as a very, very great (infinite) source of power – infinite sources! Considering that one definition of infinity is that it is a number greater than the greatest possible imaginable number, would it be likely to find or devise an infinite force and an infinite power source?

Question 2.1: Find the response (in terms of its acceleration, velocity, a...

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