Question

Use \(a(t)=-9.8\) meters per second per second as the acceleration due to gravity. Show that the height above the ground of an object thrown upward from a point \(s_{0}\) meters above the ground with an initial velocity of \(v_{0}\) meters per second is given by the function \(f(t)=-4.9 t^{2}+v_{0} t+s_{0}\)

Short answer

The height above the ground of an object thrown upward from a point \(s_{0}\) meters with an initial velocity of \(v_{0}\) meters per second is given by the function \(f(t) = s_{0} + v_{0}t - 4.9t^{2}\).

Step-by-step solution

Identify the variables

From the problem, we have \(u = v_{0}\) (initial velocity), \(g = -9.8 m/s^{2}\) (acceleration due to gravity), and \(s_{0}\) (initial height above the ground). We need to find the height above the ground \(s\) at time \(t\), which is the displacement.

Substitute into equation of motion

Substitute the given values into the equation. Since the initial height is \(s_{0}\), we can write the equation as \(s = s_{0} + ut - \frac{1}{2}gt^{2}\). When we substitute the given values into this equation, we get \(s = s_{0} + v_{0}t - \frac{1}{2}(9.8)t^{2}\).

Simplify the equation

After substitution, simplify the equation to get the height above the ground as a function of time. The simplified equation becomes \(s = s_{0} + v_{0}t - 4.9t^{2}\).

Key concepts

Acceleration Due to Gravity

When discussing motion, it's important to understand the ever-present role of gravity on Earth. Acceleration due to gravity is a constant force that pulls objects towards the Earth's center. This constant is usually represented by the symbol 'g', with a value of approximately -9.8 meters per second squared (-9.8 m/s^{2}). This acceleration value is negative because it acts in the downward direction, opposite to the initial upward motion of objects.
Imagine a ball thrown straight up; it slows down as it rises, stops momentarily at its peak, then accelerates downward, all due to gravity's influence. For any freely falling object near the Earth's surface, regardless of its mass, 'g' provides a convenient way to predict its motion, and it is crucial for calculating the object's velocity, position, and total time in the air. When calculating the motion of objects in free fall or thrown upwards, knowing how to incorporate acceleration due to gravity is fundamental in physics.

Initial Velocity

The term 'initial velocity' refers to the speed and direction an object has when it first begins moving. In physics problems, we represent initial velocity with the symbol (v_{0}) or simply 'u'. It's an essential component in equations of motion because it serves as the starting point for analyzing an object's behavior over time.
For the object thrown upward in our example, initial velocity accounts for the positive push away from the Earth. It's important to consider the direction when working with initial velocity; for upward motion, the value for v_{0} is positive, reflecting a move away from the ground. In equations of motion, the initial velocity helps us calculate kinetic energy, momentum, and displacement. It can also affect the maximum height reached by an object in the air and the total duration of flight.

Displacement

Displacement is a vector quantity that measures the shortest distance from the initial to the final position of an object, along with the direction of that straight path. In projectile motion problems, like the one involving an object thrown upwards, displacement would indicate how high above or below the initial launch position the object is at any given moment in time.
In our exercise, the displacement is represented as 's', which changes over time due to the object's initial velocity and the acceleration due to gravity. The formula incorporates both of these inputs, reflecting the fact that displacement is affected by how fast the object is moving initially and how the force of gravity influences its motion thereafter. This calculation allows us to predict where an object will be at any point during its trajectory, critical for applications ranging from ballistics to sports science.