Projectile motion - determining initial velocity and angle

In summary, the archer needs to launch his arrow at an angle of ##\alpha## and with an initial velocity of ##v_0## in order to hit the center of the target perpendicular to its surface.
  • #1
marksyncm
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Homework Statement



upload_2018-10-28_8-21-10.png


An archer launches an arrow from coordinates ##(0, 0)## at an angle ##\alpha## and with an initial velocity ##v_0##. There's a target located ahead of the archer and the center of that target is at coordinates ##(d, h)##. At what ##v_0## and at what angle ##\alpha## does the archer need to launch his arrow to hit the center of the target perpendicular to its surface?

Homework Equations



Kinematic equations

The Attempt at a Solution



We have that:

##v_x = v_0\cos(\alpha)##
##v_y = v_0\sin(\alpha) - gt##

For the arrow to hit the center of the target perpendicular to its surface, it must have a vertical velocity of zero when it reaches the target. So the following conditions must be met:

1) By the time the arrow travels a distance ##d## in the ##x## direction, its vertical velocity component should be equal to zero.
2) By that same time above, the arrow should be located at height = ##h##.

So we have:

##d = v_0\cos(\alpha)t_i \rightarrow t_i = \frac{d}{v_0\cos(\alpha)}## ........ (1)
##0 = v_0\sin(\alpha) - gt_i \rightarrow t_i = \frac{v_0\sin(\alpha)}{g}## ...... (2)
##\frac{d}{v_0\cos(\alpha)} = \frac{v_0\sin(\alpha)}{g} \rightarrow v_0^2\sin(\alpha)\cos(\alpha) = dg## .... (3)

Where ##t_i## is the time of impact.

I'm at a loss as to how to proceed from here. I'm assuming I need to use the displacement equation in the vertical direction: ##h = v_0\sin(\alpha)t - \frac{g}{2}t^2##, but I am not sure what to put into this equation. Do I put in the ##t_i## from equation (1) or from equation (2)? Can I use both values of ##t_i## simultaneously (insert one under ##t## and another under ##t^2##) since they are supposed to be the same thing? I tried doing this and I get an algebraic expression that I'm unable to process. For example, here's what happens when I use ##t_i## from equation (1):

$$h=\frac{v_0\sin(\alpha)d}{v_0\cos(\alpha)} - \frac{gd^2}{2v_0^2\cos^2(\alpha)} \rightarrow h = \frac{sin(\alpha)}{\cos(\alpha)} - \frac{gd^2}{2v_0^2cos^2(\alpha)}$$

But this equation doesn't account for the fact that at time ##t_i##, the vertical velocity needs to be zero. How do I proceed?
 

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  • #2
Your image suggests that the target is a sphere, but your solution and the formulation suggests that it is a vertical surface. I am going to assume the latter.

The value of ##t_i## must be the same in both cases. If not, your conditions would not be satisfied. However, I would suggest a different approach than using ##v_0## and ##\alpha## from the beginning. Instead, I would use the initial ##x## and ##y## components of the velocity (call them ##v_{0x}## and ##v_{0y}## for example). Either way, you should have a sufficient number of conditions to solve for all of your variables. In principle, you have three conditions and three unknowns:
  1. The vertical velocity needs to be zero at ##t_i##.
  2. The vertical distance needs to be ##h## at ##t_i##.
  3. The horizontal distance needs to be ##d## at ##t_i##.
This is a solvable system of equations.
 
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  • #3
Your image suggests that the target is a sphere, but your solution and the formulation suggests that it is a vertical surface. I am going to assume the latter.

Sorry for the confusion. Your assumption is correct.

Thanks, I'll take another stab at the problem using your approach and will see how it goes.
 
  • #4
Here's what I did:

Time when vertical velocity = 0:

$$0 = v_{0y} - gt_i \rightarrow t_i = \frac{v_{0y}}{g}$$

Vertical distance is ##h## at time = ##t_i##:

$$h = v_{0y}t - \frac{g}{2}t^2 \rightarrow H=v_{0y}\frac{v_{0y}}{g} - \frac{g}{2}\frac{v_{0y}^2}{g^2} = \frac{v_{0y}^2}{g}-\frac{v_{0y}^2}{2g} = \frac{v_{0y}^2}{2g}=h$$

Horizontal distance is ##d## at time = ##t_i##:

$$d=v_{0x}t \rightarrow d=v_{0x}\frac{v_{0y}}{g} \rightarrow d=\frac{v_{0x}v_{0y}}{g}$$

So we have two equations:

##h = \frac{v_{0}^2 sin^2(\alpha)}{2g}## ...... (1)
##d=\frac{v_{0}^2 \sin \alpha cos \alpha}{g}## ...... (2)

From equation (2), we get that ##v_{0}^2 = \frac{dg}{\sin \alpha \cos \alpha}##. Substituting this into equation (2):

$$h = \frac{dg \sin^2(\alpha)}{2g\sin(\alpha)\cos(\alpha)} = \frac{d\sin(\alpha)}{2\cos(\alpha)} = \frac{d}{2} \tan(\alpha) = h \rightarrow \frac{2h}{d} = \tan(\alpha) \iff \alpha = tan^{-1}(\frac{2h}{d})$$

Is this correct?

However, I'm not sure how to get the value for ##v_0## from equations (1) and (2). It seems that whatever substitution I make ends up with a ##v_0^2## in the numerator canceling out with another ##v_0## in the denominator. Would appreciate a tip.
 
  • #5
Well, ##\alpha## is now known so you can just substitute it into either (1) or (2) and solve for ##v_0##.

Edit: ... or you could just solve for ##v_{0x}## and ##v_{0y}## and apply Pythagoras' theorem.
 
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  • #6
Orodruin said:
Well, ##\alpha## is now known so you can just substitute it into either (1) or (2) and solve for ##v_0##.

Edit: ... or you could just solve for ##v_{0x}## and ##v_{0y}## and apply Pythagoras' theorem.

Thank you!

Is it "unusual" that I cannot find the value of ##v_0## from just equations (1) and (2) (without substituting in ##\alpha##)? Or am I just not seeing a way to do it?
 
  • #7
No, it is not unusual in any way because ##v_0## has contributions from both the vertical and the horizontal velocity components.
 
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  • #8
marksyncm said:
Thank you!

Is it "unusual" that I cannot find the value of ##v_0## from just equations (1) and (2) (without substituting in ##\alpha##)? Or am I just not seeing a way to do it?
You can use the trigonometric identity ##\sin^2(α)=\frac{\tan^2(α)}{1+\tan^2(α)}## in equation (1).
Or you can apply the double-angle formulas sin2(α)=(1-cos(2α))/2, sin(α)cos(α)=sin(2α)/2,
isolate sin(2α) and cos(2α), square and add the squares, resulting 1. You get an equation for v02.
 
Last edited:
  • #9
I think I would have begun with conservation of energy to get the vertical velocity component:

##\frac{1}{2}v_y^2 = gh## so that ##v_y = \sqrt{2 g h}##

Then, since the time to rise to the maximum height is the same as for falling from that height,

##\frac{1}{2} g t^2 = h## so that ##t = \sqrt{2 \frac{h}{g}}##

##v_x## then follows from the time and horizontal distance. The angle follows that from ##\arctan(\frac{v_y}{v_x})##.
 

Related to Projectile motion - determining initial velocity and angle

1. What is projectile motion?

Projectile motion is the motion of an object through the air that is affected by gravity.

2. How do you determine the initial velocity of a projectile?

The initial velocity of a projectile can be determined by using the equation V0 = V * cosθ, where V is the initial velocity of the object and θ is the angle at which it is launched.

3. What is the role of angle in projectile motion?

The angle at which a projectile is launched affects its trajectory and range. The optimal angle for maximum range is 45 degrees, while a higher angle will result in a shorter range and a lower angle will result in a higher range.

4. How does air resistance affect projectile motion?

Air resistance, also known as drag, can affect the motion of a projectile by slowing it down and altering its trajectory. This is why the equations for projectile motion assume a vacuum, as air resistance can complicate the calculations.

5. What are some real-life applications of projectile motion?

Projectile motion is used in a variety of fields, including sports like basketball and football, where players need to calculate the angle and initial velocity of their throw or kick to make a successful shot. It is also used in engineering and physics experiments to study the effects of gravity and air resistance on different objects.

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