Question

Velocity of a Moving Baseball A baseball is thrown upward from a height of 3 meters above Earth's surface with an initial velocity of \(10 \mathrm{~m} / \mathrm{s}\), and the only force acting on it is gravity. The ball has a mass of \(0.15 \mathrm{~kg}\) at Earth's surface a. Find the velocity \(v(t)\) of the baseball at time \(t\). b. What is its velocity after 2 seconds?

Short answer

The velocity function is \(v(t) = 10 - 9.81t\) and after 2 seconds, the velocity is \(-9.62 \, \text{m/s}\).

Step-by-step solution

Understand the problem context

The baseball is thrown upward with an initial velocity, and we need to calculate its velocity at a given time considering only the effect of gravity acting downwards. Gravity will constantly decelerate the upward motion and eventually accelerate the baseball downward.

Identify the relevant physical laws

Use the equations of motion under constant acceleration. For a body with initial velocity \(v_0\) and subject to acceleration \(a\):\[ v(t) = v_0 + at \]where \(v_0 = 10 \, \mathrm{m/s}\) and \(a = -9.81 \, \mathrm{m/s^2}\) (gravity, acting downward).

Calculate velocity as a function of time

Substitute the initial conditions into the equation of motion:\[ v(t) = 10 - 9.81t \] This equation describes the velocity of the baseball \(v(t)\) at any time \(t\).

Calculate velocity at 2 seconds

To find the velocity after 2 seconds, substitute \(t = 2\) into the velocity function:\[ v(2) = 10 - 9.81 imes 2 = 10 - 19.62 = -9.62 \, \mathrm{m/s} \] The negative sign indicates that the baseball is moving downward after 2 seconds.

Key concepts

Understanding Initial Velocity

Initial velocity is the speed at which an object starts its journey. It's a crucial component in predicting the future motion of the object. For the baseball in the exercise, the initial velocity is given as \(10 \, \mathrm{m/s}\) upwards.
That means at the moment the baseball is thrown, it instantly begins its upward journey at this speed.
In physics problems like these, initial velocity can often be a vector quantity, meaning it has both magnitude (how fast) and direction (which way).

• In this problem, since the baseball is thrown upward, the initial velocity is positive.
• If it were thrown downward, we'd consider the initial velocity to be negative.

This initial speed plays a critical role in determining how the baseball will move afterwards, particularly how high it will rise and when it'll start coming back down.

Exploring Constant Acceleration

Constant acceleration means that the rate of change of velocity is uniform over time. In the exercise scenario, gravity provides a constant acceleration to the baseball, pulling it towards the Earth.
The value of this gravitational acceleration is approximately \(-9.81 \, \mathrm{m/s^2}\), indicating a constant downward force.
This constant rate of acceleration means that the velocity of the baseball changes linearly with time.

• The negative sign in the acceleration indicates the direction—opposite of the initial upward velocity.
• Thus, as time increases, the baseball's upward velocity decreases until it reaches a peak height and begins falling back to Earth.

The laws of motion with constant acceleration are fundamental tools in physics that help us predict how an object will move when subjected to known forces.

Velocity as a Function of Time

Velocity as a function of time provides a complete description of how the speed of the baseball changes as it moves. The equation derived in the step-by-step solution is \(v(t) = 10 - 9.81t\).
This shows how the baseball's initial velocity is modified by the constant acceleration due to gravity over time.
To compute the velocity at a specific time, we simply plug in the time, \(t\), into the equation.

• This equation tells us that the baseball’s velocity decreases by \(9.81 \, \mathrm{m/s}\) every second due to gravity.
• For example, after 2 seconds, the velocity is \(-9.62 \, \mathrm{m/s}\), meaning the baseball is now moving downward.

Understanding how to calculate velocity as a function of time is crucial for analyzing motion where acceleration is constant, like the motion of a ball in free fall.