MHB What is the velocity of the jet?

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The discussion revolves around calculating the velocity of a jet given that a propeller plane travels the same distance at one-third the speed of the jet and takes 10 hours longer for the trip. The formula for the jet's velocity is established as v_jet = 3000/t_jet, while the propeller plane's time is expressed as t_prop = t_jet + 10. By equating the travel times, the relationship leads to the equation 3000/v_jet = 9000/v_jet - 10. Solving this equation reveals that the jet's velocity can be determined numerically. Ultimately, the problem emphasizes the importance of using both planes' speed and time relationships to find the jet's velocity.
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A propeller plane and a jet travel 3000miles. The velocity of the plane is 1/3 the velocity of the jet. It takes the prop plane 10 hours longer to complete the trip. What is the velocity of the jet.

Let \(\mathbf{v} = \frac{dx}{dt}\) and \(dx = 3000\). The \(dt = t - t_0\). We can always let \(t_0 = 0\) so \(\mathbf{v}_{\text{jet}} = \frac{3000}{t_{\text{jet}}}\). It appears that the information about the prop plane is unnecessary or can I use that information to determine \(t_{\text{jet}}\)?

We know that \(\mathbf{v}_{\text{prop}} = \frac{1}{3}\mathbf{v}_{\text{jet}}\) and the time require is \(t_{\text{prop}} = t_{\text{jet}} + 10\).
 
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Letting the velocity of the jet be $v$ and observing that both planes travel the same distance $d$, we may write:

$$d=vt=\frac{v}{3}(t+10)$$

For $0<d$, we must have $0<v$, and so we may divide through by $v$ to obtain:

$$t=\frac{t+10}{3}\implies t=5$$

Hence:

$$v=\frac{d}{5}$$
 
dwsmith said:
A propeller plane and a jet travel 3000miles. The velocity of the plane is 1/3 the velocity of the jet. It takes the prop plane 10 hours longer to complete the trip. What is the velocity of the jet.

Let \(\mathbf{v} = \frac{dx}{dt}\) and \(dx = 3000\). The \(dt = t - t_0\). We can always let \(t_0 = 0\) so \(\mathbf{v}_{\text{jet}} = \frac{3000}{t_{\text{jet}}}\). It appears that the information about the prop plane is unnecessary or can I use that information to determine \(t_{\text{jet}}\)?
"v_{\text{jet}}= \frac{3000}{t_{text}} is a formula for the speed of the jet. This problem is asking for a specific numerical answer.

We know that \(\mathbf{v}_{\text{prop}} = \frac{1}{3}\mathbf{v}_{\text{jet}}\) and the time require is \(t_{\text{prop}} = t_{\text{jet}} + 10\).
I would write, rather, that the time is \frac{3000}{v_{\text{jet}}}. The time required for the prop plane to fly 3000 mi would be \frac{3000}{v_{\text{prop}}}= \frac{3000}{\frac{1}{3}v_{\text{jet}}}= \frac{9000}{v_{\text{jet}}} and that is 10 hours more than the time required for the jet:
\frac{3000}{v_{\text{jet}}}= \frac{9000}{v_{\text{jet}}}- 10.

Solve that equation.
 
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