MHB What is the velocity of the jet?

[Dustinsfl]

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 \(t_{\text{prop}} = t_{\text{jet}} + 10\).

 

[MarkFL]

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:

$$v=\frac{d}{5}$$

 

[HallsofIvy]

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.