Question

You throw two objects with differing masses \(m_{1}\) and \(m_{2}\) upward into the air with the same initial velocity \(a \mathrm{ft} / \mathrm{s}\). What is the difference in their velocity after 1 second?

Short answer

The difference in their velocities after 1 second is 0.

Step-by-step solution

Understand the Problem

We are given two objects with masses \(m_1\) and \(m_2\), both thrown upward with the same initial velocity \(a\). We need to determine the difference in their velocities after 1 second.

Identify Key Concepts

The velocity of an object thrown upwards is affected by gravity, which slows it down. The velocity after some time \(t\) can be calculated using the formula \(v(t) = a - g \times t\), where \(g\) is the acceleration due to gravity (32 \(\text{ft/s}^2\) on Earth).

Calculate the Velocity After 1 Second

Using the formula \(v(t) = a - g \times t\), substitute \(t = 1\) second and \(g = 32\,\text{ft/s}^2\). Thus, the velocity after 1 second for both objects is \(v = a - 32 \times 1 = a - 32\).

Calculate the Difference in Velocities

Since both objects are thrown with the same initial velocity and are affected by gravity in the same way, their velocities at any point in time, including after 1 second, will be equal. Therefore, the difference in their velocities is \(v_1(t) - v_2(t) = (a - 32) - (a - 32) = 0\).

Key concepts

Velocity Calculation

When we calculate velocity, we consider both the direction and the speed at which an object moves. Velocity is different from speed because it includes both magnitude and direction. If an object is thrown upwards, it has an initial upward velocity but gravity will gradually reduce it.
To calculate the velocity of an object after a certain time, we need to adjust for gravity, which always pulls objects towards the Earth. The basic formula to find the velocity at time \( t \) is:

• \( v(t) = a - g \times t \)

Where:

• \( v(t) \) is the velocity at time \( t \)
• \( a \) is the initial velocity or the speed the object starts with
• \( g \) is the acceleration due to gravity
• \( t \) is the time that has passed

Plug in values for \( t \) and \( g \) to get the velocity after any amount of time.

Acceleration Due to Gravity

Gravity is a force that every object with mass exerts on another. On Earth, gravity gives objects an acceleration when they fall. This is known as "acceleration due to gravity." Its value is about \( 32 \text{ ft/s}^2 \) (or approximately \( 9.81 \text{ m/s}^2 \) in metric units).
Whenever an object is thrown upward, the gravity acts against the motion, slowing it down. This is why, after some time, an object thrown upwards eventually starts to fall back down. When we calculate how the velocity of an object changes over time, gravity is a crucial factor.
It's important to remember that acceleration due to gravity is constant near the Earth's surface. Hence, when calculating velocities in situations like these, we assume this value stays the same throughout the motion. This makes predictions about velocity much simpler.

Initial Velocity

Initial velocity is the speed at which an object starts its motion. It's effectively the 'starting push' given to the object. In our context, it's the speed with which the objects were thrown into the air. Initial velocity is crucial as it sets the stage for all subsequent motion.
Denoted by \( a \) in the example exercise, initial velocity determines how high or far the object will travel before gravity slows it down completely and brings it back to the ground. Even if two objects have the same initial velocity, their subsequent paths might vary based on other factors like mass or air resistance.
However, in a vacuum, or where additional forces like air resistance are not considered, every object with the same initial velocity thrown upwards will behave identically. This is why in the exercise, objects with different masses but the same initial velocity ended up at the same velocity after one second.