# Projectile motion of an arrow to an apple

• ornitho
In summary, the conversation discusses the well-known legend of William Tell shooting an apple off his son's head with an arrow. The question posed is what angle did William use to aim the arrow if it had an initial speed of 55m/s and the boy was 25m away. The conversation then goes on to discuss equations for velocity and position and how to use them to solve for the unknowns.
ornitho
im having a very difficult time with this question:

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 25m away, at what launch angle did William aim the arrow? The arrow and apple are initially at the same height above the ground.

$$x = v_{0}\cos \theta \; t$$
$$y = v_{0}\sin \theta\; t -\frac{1}{2}gt^{2}$$

$$v_{0} = 55$$So you have:

$$25 = 55\cos \theta \; t$$
$$0 = 55\sin \theta\; t - 5t^{2}$$

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I do not have time value though :(

y = 0 wouldn't it, as it is returning to the initial y position, but i still do not understand how I am supposed to find t

The maximum altitude occurs at $$t = \frac{v_{0}\sin\theta}{g}$$.

The maximum altitude is: $$\frac{v_{0}^{2}}{2g}\sin^{2}\theta$$

ornitho said:
y = 0 wouldn't it, as it is returning to the initial y position, but i still do not understand how I am supposed to find t

That's right! now if you understand courtrigrad's earlier equations, for velocity and position, you have two equations, and two unknowns (t and the angle, theta). You can then solve the set of equations for both unknowns.

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## What is projectile motion?

Projectile motion is the motion of an object through the air under the force of gravity. It follows a curved path called a parabola.

## How does the projectile motion of an arrow to an apple work?

The arrow is launched with an initial velocity and then follows a parabolic path towards the apple. The force of gravity acts on the arrow, causing it to accelerate downwards, while the initial velocity keeps it moving horizontally. The arrow will eventually hit the apple if launched at the correct angle and velocity.

## What factors affect the projectile motion of an arrow to an apple?

The factors that affect projectile motion include the initial velocity, angle of launch, air resistance, and the mass and shape of the projectile. These factors can be manipulated to achieve a desired trajectory.

## How is the angle of launch determined for accurate projectile motion?

The angle of launch is determined using trigonometric functions and the desired range and height of the projectile. The optimal angle for maximum distance is 45 degrees, but for hitting a target at a specific height, the angle may need to be adjusted.

## What real-life applications does projectile motion have?

Projectile motion is used in a variety of real-life applications, such as sports (e.g. throwing a ball, shooting a basketball), military operations (e.g. calculating the trajectory of a missile), and even in amusement park rides (e.g. roller coasters). Understanding projectile motion is also important in fields such as engineering and physics.

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