Projectile Motion test problem, involving unk. v0 and y displacement.

[prohme]

Homework Statement

A movie set requires a stunt man to jump across a 40m wide river on a motorcycle. The takeoff ramp on one bank of the river is inclined at 53(degrees) above the horizontal and is 100m above the level of the river. If the bank on the far side is 15 m below the top of the takeoff ramp, what minimum velocity must he have to reach the opposite bank?

Homework Equations

$$\Delta$$y = v_0yt - .5gt²
$$\Delta$$x = v_0xt
R = [tex]\frac{v²sin2\Theta}{g}[/tex]

The Attempt at a Solution

I've made several attempts to solve this during the test, I'm completely stuck though.
With having 2 variables missing I can't solve the conventional way (Solving for t using the y displacement) because I don't know the 'y' component of velocity.

I don't see how it is possible to solving for t in the x dimension because, ini x velocity is unknown and time is unknown.

Using the max range formula wouldn't do me any good either, because there is a displacement in the y dimension. Even if I try to partially solve using the max range formula I fail, I do not know what distance before the cliff to use as range R
-----------------------------------------------------------------------------

Right now I'm thinking this problem isn't possible to solve, which is most likely not the case considering it was a test problem.

Extremely eager for someone to prove me wrong:
All help appreciated
-Pat

 

[Delphi51]

I think you can do it. Just pick a letter like v for the initial speed and "pretend" you know it so you can write out the separate equations for the horizontal and vertical motion. Some people just can't handle an unknown v and have to run all that through with some number like 47 in place of the v, then erase all the 47's and put in v's afterward!

Anyway, I think you will end up with one horizontal equation and two vertical equations (velocity and distance) with unknowns v and t. You should then be able to solve two of the equations as a system to find v.

 

[prohme]

Let me know if anyone else has any other solutions.

I don't think that we had ever solved any problems as a system of equations. I tried doing this problem thoroughly as a substitution, but it got really messy.

 

[Delphi51]

You might post your work so we can see what went wrong.
Substitution is one method of solve a system of equations.
We aren't supposed to provide a solution; just help you with yours.

 

[rwisz]

rick

I would first like to commend you for your attempt to solve the problem on your own. It shows your critical thinking skills and determination to find a solution. However, I do believe that this problem is possible to solve.

First, let's break down the problem into smaller parts. We know that the stunt man needs to jump 40m horizontally and the takeoff ramp is 100m above the river. This means that the total distance he needs to travel in the air is 100m + 40m = 140m. We also know that the bank on the far side is 15m below the top of the takeoff ramp. This means that the vertical displacement is 100m - 15m = 85m.

Now, let's focus on the x direction first. We can use the equation \Deltax = v0xt to solve for the initial velocity in the x direction. We know that the displacement in the x direction is 40m and the time of flight is the same for both the x and y directions. Therefore, we can write the equation as 40m = v0x(t). But we still have two unknowns, v0x and t.

Next, let's look at the y direction. We can use the equation \Deltay = v0yt - .5gt2 to solve for the initial velocity in the y direction. We know that the displacement in the y direction is 85m and the time of flight is the same for both the x and y directions. Therefore, we can write the equation as 85m = v0y(t) - .5gt2. Again, we have two unknowns, v0y and t.

Now, we have two equations with two unknowns. We can solve for t in one equation and substitute it into the other equation to solve for v0x or v0y. Since we know that the angle of inclination is 53(degrees), we can use the equation R = \frac{v2sin2\Theta}{g} to solve for t. We know that R = 140m, g = 9.8m/s2 and \Theta = 53(degrees). Substituting these values into the equation, we get t = 5.05 seconds.

Now, we can substitute this value of t into either of the previous equations to solve for v0x or v