Question

An arrow is shot straight up into the air and on its return strikes the ground at \(260 \mathrm{ft} / \mathrm{s}\), imbedding itself \(9.0\) in. into the ground. Find \((a)\) the acceleration (assumed constant) required to stop the arrow, and \((b)\) the time required for the ground to bring it to rest.

Short answer

The acceleration required to stop the arrow is approximately \( -90267 \mathrm{ft} / \mathrm{s}^2\), and the time required to bring it to rest is approximately \(0.00288\) seconds.

Step-by-step solution

Identify given values

Here the final velocity (\(v_f\)) is zero because the arrow ultimately comes to rest. The initial velocity (\(v_i\)) is \(260 \mathrm{ft} / \mathrm{s}\) (the speed at which the arrow hits the ground). And the displacement (\(d\)) is \(9.0\) inches (which needs to be converted to feet, giving \(d = 0.75\) feet).

Find acceleration

To find the acceleration we apply the formula \(v_f^2 = v_i^2 + 2ad\) where \(a\) is acceleration. By rearranging the formula, we can solve for \(a\): \(a = (v_f^2 - v_i^2) / (2d) \). Substituting the given values into the equation and solving gives \( a = (0 - (260)^2) / (2*0.75) \), resulting in \(a \approx -90267 \mathrm{ft} / \mathrm{s}^2 \).

Find time

To determine the time (\(t\)) required to bring the arrow to rest, we use the equation \(v_f = v_i + at\). Rearranging to solve for \(t\), we get \(t = (v_f - v_i) / a \). Substituting the values in, we get \(t = (0 - 260) / -90267\), so \(t = 260 / 90267 \approx 0.00288\) seconds.

Key concepts

Final Velocity

In kinematic problems, the term 'final velocity' represents the speed and direction of a moving object at the conclusion of a considered time interval. For example, when an arrow hits the ground and then comes to a full stop, like in the given exercise, its final velocity (\(v_f\)) is zero because it is no longer in motion. This is an essential part of understanding motion, as it provides valuable information about the journey of an object from start to end.

It's important to note that final velocity can be affected by various factors including acceleration, initial velocity, and the time for which these factors are applied. In the context of our problem, final velocity is crucial for calculating the necessary acceleration and time to bring the arrow to rest, as it represents the moment of the arrow's motion where these calculations are aimed.

Initial Velocity

The concept of 'initial velocity' refers to the speed and direction of an object at the start of a time interval. In kinematic equations, the initial velocity is often denoted by \(v_i\). For our arrow, the initial velocity is the speed at which it hits the ground, which is 260 feet per second. This concept is vital as it serves as a starting point for calculations involving an object's motion.

Understanding initial velocity helps students to set up other kinematic calculations such as predicting the final velocity or displacement after a certain period of time, given constant acceleration. Remember, initial velocity, along with final velocity, is used to determine the acceleration needed to change the speed of an object over a specific distance.

Constant Acceleration

Constant acceleration is a common assumption in kinematic problems, simplifying the calculations needed to understand an object's motion. It implies that the acceleration, the rate of change of velocity, remains unchanged throughout the duration of the object's journey. In our exercise, the assumption of constant acceleration allows for the use of specific motion equations to determine the arrow's behavior as it strikes the ground and embeds itself into the soil.

By using the formula \(v_f^2 = v_i^2 + 2ad\), where \(a\) stands for acceleration, we can calculate the negative acceleration exerted by the ground to stop the arrow. This calculated acceleration is a large negative value, indicating a rapid deceleration over the short distance of embedding into the ground, which is consistent with the concept of constant acceleration over the stopping distance.

Motion Equations

Motion equations are mathematical formulas used to describe the relationships between displacement, velocity, acceleration, and time. They are also known as the ‘equations of motion’ and are fundamental in solving kinematic problems. There are several key equations, but the two used in this exercise are \(v_f^2 = v_i^2 + 2ad\) for acceleration and \(v_f = v_i + at\) for time.

By rearranging these equations, as shown in the step-by-step solution, you can isolate and calculate for unknown variables. These equations provide the means to understand how different factors of motion relate to one another. It's useful to memorize these equations for solving various kinematics problems, as they often form the backbone of the solutions.