Projectile Motion: Finding Initial Velocity with Given Height and Speed

[v1ru5]

Homework Statement

A projectile is fired with an initial speed v at an angle 40 degrees above the horizontal from a height of 40m above the ground. The projectile strikes the ground with a speed of 1.4v. Find v.

The Attempt at a Solution

All I could find i Vinitialx=vcos40 and Vinitialy=vsin40 and everytime I try to use any equation, time is always an issue since it is not given. Any help would be appreciated.

 

[Simon Bridge]

Hint: simultaneous equations

Generally, ballistic (projectile motion) problems involve initial and final heights which are not the same so the usual sets of ballistic equations won't work without some fiddling. You are better served by using v-t diagrams for each component and constructing the equations.

 

[v1ru5]

Here are some projectile equations I have:

1) Vx = Vix (since no acceleration in x direction)
2) x = xi + vix*t (not useful since I do not know the displacement in the x direction)
3) Vy = Viy - g*t (cannot use this since I do not know t)
4) y = yi + Viy*t - 1/2*gt^2 (not useful since I do not know t)

So since I already know what Vx is, all I need to do now is find Vy. I can use equation 4) to find t and sub it into equation 3) but then I will still have two unknowns which would be Vy and V so I am really confused. Any help would be great! Thanks.

 

[Simon Bridge]

You left off:
5) V^2 = Vx^2 + Vy^2

List what you know.
$\theta=40^\circ$ and $x_i=y_i=0,\; x_f=x(T),\; y_f=y(T)=-40m,\; g=9.8m/s^2,\; v_{xf}=v_x(T)=v_{xi},\; v_{yf}=v_y(T)$
... (T is the time to hit the ground) rewrite the equations 1-5.

Can you use equations 3 and 4 as simultanious equations to eliminate T?

I see you know that $v_f=1.4v$
Thinking outside the box: would conservation of energy work for this?

 

[jaumzaum]

you don't need any position x time or torricelli equation to solve this, its only a energy conservation problem

Mv²/2 + MgH = constant

 

[Simon Bridge]

You are in luck - jaumzaum just gave you the answer.
We do not normally do this for homework questions - it is usually considered bad form. It is better for your understanding if you figure these things out yourself which is why you get more round-about answers sometimes.

 

[v1ru5]

You left off:
5) V^2 = Vx^2 + Vy^2

List what you know.
$\theta=40^\circ$ and $x_i=y_i=0,\; x_f=x(T),\; y_f=y(T)=-40m,\; g=9.8m/s^2,\; v_{xf}=v_x(T)=v_{xi},\; v_{yf}=v_y(T)$
... (T is the time to hit the ground) rewrite the equations 1-5.

Can you use equations 3 and 4 as simultanious equations to eliminate T?

I see you know that $v_f=1.4v$
Thinking outside the box: would conservation of energy work for this?

I am still a little confused and we haven't gotten to the energy unit yet so I don't think it has anything to do with that. I was also thinking of using the equation Vf^2=Vi^2+2ad so I would get something like (1.4v)^2=(vsin40)^2+2*9.8*40. I am not sure if I am doing that correctly. I agree with you that you should not just give me the answer, but rather help me understand the question so I can find the solution myself. Thanks for your help!

 

[Simon Bridge]

Well, if I were you, I'd use the conservation of energy method anyway (you've heard of it right?) ... it is fast and would get you bonus marks. If you are worried that the method is not right for the unit you are doing, you can still do it the long way as well ... the answers should agree.

For the long way:
6. $v_{yf}^2=v_{yi}^2+2ad$
... well done - that is what the algebra before would have got you to.
(When you try the conservation of energy method - compare the equation you get with this one.)

To use the equation approach -
1. separate into components
2. list what you know and what you need to know
3. list the kinematic equations
4. pick the equation that has what you need to know as well as what you know.
(you may need to use some other relationship given to you as well.)

5. rearrange to make the needed quantity the subject.

You are at step 5.

In the long run, it is more useful to get comfortable with the algebra.