Motion along a curved path -- introductory

• kkdj20
In summary, to solve for the launch angle of William Tell's arrow, you need to use the equations for projectile motion and set up a system of equations with two variables. Then, solve for either time or theta and substitute it into the other equation to solve for the launch angle. It is recommended to use the substitution method to solve for theta, which involves factoring or using the quadratic formula.
kkdj20
1. "William Tell is said to have shot an apple off his son's head with an arrow. If the arrow was shot with an initial speed of 55m/s and the boy was 15m away, at what launch angle did Bill aim the Arrow? (Assume that the arrow and the apple are initially at the same height above the ground.)"

The Attempt at a Solution

Change in X is 55m/s*cos(theta)*t
Change in Y is 55m/s*sin(theta)*t - (4.9*t^2)
Need to solve for either time or theta in order to get the other from what I can see? Is there another way? Or am I wrong all together?

Didn't mention, I don't need the answer; I have it. I just need a reliable way to solve it. A very in depth-step-by-step with explanations would be very beneficial to me, thank you. And my teacher is also not looking for an answer, so don't fret about "cheating," the assignment was given in order to further understanding of the topic and to be able to solve more similar problems in the future.

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It helps if you write regular equations to describe the motion of the arrow.

If the arrow is shot at an initial angle, its altitude will increase until gravity arrests its upward motion. Once the upward motion of the arrow ceases, it's going to start to fall back to earth. Simultaneously, the arrow is traveling at constant horizontal velocity towards the target.

By writing equations which describe this motion, you should be able to solve for both the time it takes the arrow to reach the target and the angle at which it is shot.

That honestly doesn't help me at all, could you please show some step by step solving? If you need to use a problem with similar goals and different numbers that's fine, but at the moment I'm no better off

I know those equations, but you need theta to solve for time and vise-versa,
my question is how to use that information to be able to solve for theta or time.

##\Delta X = (55 m/s) \times \cos(\theta)t##
##\Delta Y = (55 m/s) \times \sin(\theta)t - (4.9t^{2}) ##

This is a system of two equations with two variables, you solve through substitution or any other algebraic way of solving systems of equations. One such way is solving for t in the first equation:

## t = \frac{\Delta X}{(55 m/s)\cos(\theta)} ##

And plugging it into the second equation:

##\Delta Y = (55 m/s) \sin(\theta)\frac{\Delta X}{(55 m/s)\cos(\theta)} - (4.9(\frac{\Delta X}{(55 m/s)\cos(\theta)})^{2}) ##

And work from there, solving for ##\theta##, but this way is rather messy. I would suggest doing it using the other substitution, (what is ##\Delta Y##? and how can you solve for t (think factoring or quadratic formula) in terms of ##\sin(\theta)##?). Plugging it into the first equation, you should get some equation with either the cotangent or tangent of the angle equaling some number and you can take the inverse cotangent or inverse tangent of that number.

Hope this helps.

Thank you , that actually did help, but don't say "plugging it in", it's substitution :p anyways thank you; they oughta make you a helper person, not King ambiguous Steam

1. What is motion along a curved path?

Motion along a curved path refers to the movement of an object along a curved trajectory, rather than in a straight line. It is a fundamental concept in physics and can be observed in many real-life scenarios, such as the motion of a ball thrown in the air or the orbit of a planet around a star.

2. What causes an object to move along a curved path?

An object moves along a curved path due to the presence of a centripetal force. This force is directed towards the center of the curve and is responsible for changing the direction of the object's motion. In the absence of a centripetal force, the object would continue to move in a straight line.

3. How is motion along a curved path different from motion in a straight line?

The main difference between motion along a curved path and motion in a straight line is the presence of a centripetal force. In motion along a curved path, the direction of the object's velocity is constantly changing, while in motion in a straight line, the direction of the object's velocity remains constant.

4. What is the relationship between speed and velocity in motion along a curved path?

Speed and velocity are closely related in motion along a curved path. While speed refers to the rate of change of distance, velocity refers to the rate of change of displacement. In motion along a curved path, the speed of an object may remain constant, while its velocity changes due to the changing direction of its motion.

5. How is motion along a curved path calculated?

Motion along a curved path can be calculated using the principles of kinematics, which involve measuring the displacement, velocity, and acceleration of the object. The centripetal force can also be calculated using the object's mass, speed, and radius of curvature.

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