Problem 37 - Technical Mathematics with Calculus

Question

The formula for the displacement \(s\) of a freely falling body having an initial velocity \(v_{0}\) and acceleration \(a\) is $$s=v_{0} t+\frac{1}{2} a t^{2}$$ Solve this equation for \(a\)

Step-by-Step Solution

Answer

The acceleration 'a' is given by the formula: $$a = \frac{2(s - v_0 t)}{t^2}$$

Step 1: Isolate the term with 'a'

Identify the term containing the variable 'a' and separate it from the rest of the equation. Start by subtracting the term containing the initial velocity, which is \(v_0 t\), from both sides of the equation to isolate the term with acceleration. This gives you: \begin{align*} s - v_0 t &= \frac{1}{2} a t^2\text{or} s - v_0 t &= \frac{1}{2} a t^2\text{which simplifies to} 2(s - v_0 t)&= a t^2\text{by multiplying both sides by 2.}\text{(Make sure to include the last step in step 2)}\text{Now all the terms containing 'a' are in one part of the equation.}

Step 2: Solve for 'a'

With the terms involving 'a' isolated, we can solve for 'a' by dividing both sides of the equation by \(t^2\). This yields: \begin{align*} \frac{2(s - v_0 t)}{t^2} &= a \text{or, simplifying further,} a &= \frac{2(s - v_0 t)}{t^2}\text{Now 'a' is isolated and the equation is solved for 'a'.}\text{Recall that dividing by \(t^2\) assumes that \(t\) is not equal to zero.}

Key Concepts

Kinematic Equations

Kinematic equations are a set of equations used to describe the motion of objects without considering the forces that cause the motion. They are particularly useful in describing the motion of objects that are moving with constant acceleration. When it comes to free fall, an object is only acted upon by gravity and hence has a constant acceleration, which on Earth's surface is approximately 9.8 m/s². The displacement equation in the problem,
\(s = v_{0}t + \frac{1}{2} at^{2}\),
is a classic example of a kinematic equation. It relates the four key kinematic quantities: displacement (\(s\)), initial velocity (\(v_{0}\)), acceleration (\(a\)), and time (\(t\)). Understanding how these variables interplay offers invaluable insight into predicting an object's future position, given certain initial conditions.

Solving for Acceleration

Solving for acceleration in kinematic equations often involves isolating the acceleration term on one side of an equation. In our exercise, we start by eliminating the initial velocity term from the equation to focus on the part containing the acceleration. After subtracting \(v_0 t\) from both sides, we multiply through by 2 to get rid of the fraction. This results in
\(2(s - v_0 t) = a t^2\).
The next step is a critical concept: in kinematics, if you're solving for one quantity, you often have to ensure that all the other quantities are known or can be cancelled out. In this case, we divide both sides by \(t^{2}\) to get acceleration by itself. Doing so leads to
\(a = \frac{2(s - v_0 t)}{t^2}\),
which is the formula for solving for acceleration given the initial velocity, displacement, and time.

Initial Velocity

The initial velocity, denoted as \(v_{0}\), is the velocity of the object at the start of the time period considered. It's an essential component of kinematic equations because it sets the stage for how the object behaves over time. In free fall problems, the initial velocity can be zero, such as when an object is dropped from rest, or it can have a value if, for example, it's thrown upwards or downwards. Knowing the initial velocity allows us to determine how much the object's speed will increase or decrease due to acceleration. In the equation provided, the term \(v_{0} t\) combines the initial velocity with time to calculate the portion of the displacement attributed to this initial motion before acceleration impacts the motion.

Displacement

Displacement refers to the change in position of an object and is a vector quantity, which means it has both magnitude and direction. In kinematics, displacement is different from distance since it only measures the straight-line change from the initial to the final position, ignoring the path taken. The kinematic equation provided in the question incorporates displacement as \(s\). This equation shows how displacement can be calculated from initial velocity, acceleration, and time. When the displacement is understood in the context of these other variables, it becomes a powerful tool in predicting an object's position after a certain period of time.