Projectile Motion: Finding a Relationship

[RedBarchetta]

Homework Statement

A projectile is launched when a car goes by t=0 s. (Position is given as meters and time as seconds). The cars has an initial velocity of 12 m/s. Find the relationship between Vo and the angle theta such that the projectile will always hit the car. When you determine the relationship, make sure to test it with a few values of Vo and theta.

Homework Equations

Vox= Vo * cos(theta of Vo)

The Attempt at a Solution

Well given that equation, since the car is traveling at 12 m/s, the projectile's x-component of the inital velocity must be equal to the car's, which will make the ball travel parallel with the car as it completes its parabolic curve.

Now I've plugged in a couple of different magnitudes to the above equation that work:

25 m/s, 61.31 degrees

20 m/s, 53.13 degrees

15 m/s, 36.87 degrees

So what would you say is the relationship? I'm lost.

 

[anathema]

Thank you for your post. The relationship between Vo and theta for the projectile to always hit the car can be determined by using the equation Vox = Vo * cos(theta of Vo). As you correctly determined, the x-component of the initial velocity must be equal to the car's velocity in order for the projectile to hit the car. This means that Vox = 12 m/s.

Now, using the equation Vox = Vo * cos(theta of Vo), we can rearrange it to solve for theta of Vo:

theta of Vo = arccos(Vox / Vo)

Substituting in the value of 12 m/s for Vox, we get:

theta of Vo = arccos(12 / Vo)

Therefore, the relationship between Vo and theta is:

Vo = 12 / cos(theta of Vo)

This relationship can be tested with different values of Vo and theta, as you have done, to ensure that the projectile will always hit the car. I hope this helps clarify the relationship between Vo and theta for you. Keep up the good work in your studies!

 

[Moetasim]

The relationship between Vo and theta is that as the initial velocity (Vo) increases, the angle (theta) at which the projectile is launched must decrease in order for the projectile to always hit the car. This can be seen in the examples provided, where as Vo increases, theta decreases. Additionally, the relationship can be expressed as an inverse relationship, where Vo and theta are inversely proportional to each other. This means that as one increases, the other decreases and vice versa. This relationship can be verified by testing more values of Vo and theta and observing the trend.