# [SOLVED]Velocity of a jet

#### dwsmith

##### Well-known member
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

Staff member
Letting the velocity of the jet be $v$ and observing that both planes travel the same distance $d$, we may write:

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

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

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

Hence:

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

#### HallsofIvy

##### Well-known member
MHB Math Helper
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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