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 velocity after 1 second is 0.

Step-by-step solution

Understand the Effect of Gravity

When objects are thrown upward, the only force acting on them in the vertical direction is gravity. Gravity acts downward, reducing the velocity of the objects by approximately 32 feet per second every second.

Identify Initial Velocities

Both objects are launched with the same initial velocity of \(a \ \mathrm{ft/s}\).

Calculate Velocity after 1 Second for Object 1

For object with mass \(m_1\), the velocity after 1 second can be calculated as: \( v_{1} = a - 32 \). This accounts for the reduction in velocity due to gravity.

Calculate Velocity after 1 Second for Object 2

Similarly, for the object with mass \(m_2\), the velocity after 1 second is: \( v_{2} = a - 32 \).

Evaluate Difference in Velocities

Since both objects have the same initial velocity and experience the same gravitational effect, their velocities after 1 second remain \( v_{1} = v_{2} = a - 32\). Thus, the difference in their velocities is \( v_{1} - v_{2} = 0 \).

Key concepts

Effect of Gravity

Gravity is a force that constantly acts on objects by pulling them toward the center of the Earth. When you throw an object up into the air, gravity works against the motion, pulling it back down. This occurs at a constant rate, leading to a deceleration of approximately 32 feet per second squared (often denoted as 32 ft/s²).

In kinematics, this force is crucial because it affects any object's motion that is moving vertically. Regardless of how fast you initially throw something, gravity will decrease its speed uniformly over time. This loss in speed is why no matter how hard you throw a ball upwards, it will eventually stop rising and fall back down. Understanding gravity is key to predicting how objects move in the real world.

Initial Velocity

Initial velocity refers to the speed and direction an object has when it first begins its motion. In our scenario, two objects are thrown upward with the same initial velocity denoted as \( a \ ft/s \).

Knowing the initial velocity is important because it sets the starting condition for any further calculations of motion. The initial velocity will determine how high the objects will climb and how long they can stay in the air before gravity brings them back down. When two objects have identical initial velocities, they start their journey in the same way, even if they have different masses.

For any problems involving motion, identifying the initial velocity is the first essential step.

Velocity Calculation

Calculating velocity after a certain period of time involves understanding the concept of change in speed due to external forces, like gravity. In this exercise, gravity reduces the speed of each object by 32 feet per second.

To calculate the velocity after one second, you subtract the impact of gravity from the initial velocity. Therefore, the velocity \( v \) after one second for any object that is thrown upwards can be expressed as:
\[ v = a - 32 \]
This formula applies to both objects regardless of their masses in this context.

• The formula helps predict future motion.
• Allows us to assess how motion changes due to gravity.

Overall, understanding how to calculate velocity accurately gives you a more complete view of how objects behave when in motion.

Mass Independence in Vertical Motion

One interesting aspect of kinematics in a gravitational field is that an object's mass does not affect its vertical motion. This concept stems from the fact that gravity accelerates all objects at the same rate, around 32 ft/s², regardless of their mass.

Thus, when two objects are thrown upwards with identical initial velocities, their final velocities after one second are the same, even though they have different masses. This means:

• Mass does not affect how much their speeds are reduced by gravity.
• The downward force acting on both is identical, highlighting that velocity changes are purely a result of gravity.
  
• This principle is a key foundation in understanding classical physics.

Recognizing mass independence in vertical motion can simplify many complex problems, focusing calculations solely on forces acting due to motion and gravity.