Question

An archer is shooting at targets. The height of the arrow is 5 feet above the ground. Due to safety rules, the archer must aim the arrow parallel to the ground.

Short answer

The arrow will hit the ground in approximately 0.56 seconds.

Step-by-step solution

Understand the Given Information

The arrow is released parallel to the ground, at a height of 5 feet. There is no initial vertical velocity since the arrow isn't launched upwards or downwards.

Determine the Appropriate Physics Law

We'll use the law of gravitation and motion to solve this. One such law, given by: \( h = ut + 0.5gt^2 \), where \( h \) is the height, \( u \) is the initial vertical speed, \( g \) is the acceleration due to gravity, and \( t \) is the time. Due to the direction of gravity, its sign will be negative.

Substitute Known Values

Substitute the known values into the equation. Here, \( h = 5 \) ft, \( u = 0 \), \( g \) is approximately \( -32 \) ft/s². The equation now reads as: 5 = 0*t +0.5*-32*t².

Solve for Time

On simplifying the equation, we get -32*t² = -10, which gives us the value of \( t = \sqrt{10/32} \). Therefore, the time taken is roughly 0.56 seconds, that's how long it takes for the arrow to hit the ground.

Key concepts

Gravity

Gravity is an invisible force that pulls objects toward the Earth. It acts as the constant downward force that affects any object in projectile motion, like the arrow in our exercise.
The force of gravity on Earth is roughly 32 feet per second squared (ft/s²). This means that for every second an object is in motion, it falls faster towards the Earth.
This acceleration is always negative when using the equation of motion since it acts downward. This is crucial in calculating how long it takes for an arrow, initially parallel to the ground, to hit the earth.

Initial Velocity

In projectile motion, the initial velocity refers to the object's speed and direction at the moment it is released.
For the archer's arrow, since it is shot parallel to the ground, the initial vertical velocity is zero.
This means that at the start, the arrow doesn't move up or down in the vertical direction; it only begins to fall once gravity takes effect.
Remember that the initial velocity impacts the overall path and range of the projectile, just not the height when shot parallel to the ground.

Equation of Motion

The equation of motion is a fundamental tool used in physics to describe the movement of objects influenced by forces like gravity.
A key equation used here is:

• \(h = ut + 0.5gt^2\)

This equation relates an object's vertical height (\(h\)), initial velocity (\(u\)), gravitational acceleration (\(g\)), and the time in the air (\(t\)).
For our arrow, substituting 0 for initial velocity shows that only gravity and time affect how the arrow falls to the ground.

Time of Flight

The time of flight is how long a projectile stays in the air before it hits the ground.
It is crucial for the archer to know because it tells how long the arrow will travel.
By solving the motion equation with the height, initial speed, and gravity, we find that the time of flight is approximately 0.56 seconds.

• This tells us how quickly gravity pulls the arrow down to earth from a 5-feet height.
• With this knowledge, predictions about the arrow's trajectory and point of impact are more accurate.