Question

Same forces act on two bodies of different mass \(2 \mathrm{~kg}\) and \(5 \mathrm{~kg}\) initially at rest. The ratio of times required to acquire same final velocity is (A) \(5: 3\) (B) \(25: 4\) (C) \(4: 25\) (D) \(2: 5\)

Short answer

The ratio of times required for the two masses to reach the same final velocity is \(2:5\). So the correct answer is (D) \(2: 5\).

Step-by-step solution

Apply Newton's second law of motion to each mass

In this step, we will set up equations for each mass. Based on Newton's second law of motion, \(F = ma\), we have: For mass \(2\mathrm{~kg}\): \(F = m_1a_1\) For mass \(5\mathrm{~kg}\): \(F = m_2a_2\) Since the same force acts on both masses, we can write: \(m_1a_1 = m_2a_2\) Now we can find the ratios of their accelerations: \(\frac{a_1}{a_2} = \frac{m_2}{m_1}\) Plug in the values of the masses: \(\frac{a_1}{a_2} = \frac{5}{2}\)

Use the equation for final velocity

To find the time required for both masses to reach the same final velocity, we can use the equation: \(v = u + at\) where \(v\) is the final velocity, \(u\) is the initial velocity, \(a\) is acceleration, and \(t\) is time. Both bodies are initially at rest, so \(u = 0\). Therefore, the equation becomes: \(v = at\) Now we can find the time for each mass: For mass \(2\mathrm{~kg}\): \(t_1 = \frac{v}{a_1}\) For mass \(5\mathrm{~kg}\): \(t_2 = \frac{v}{a_2}\)

Calculate the ratio of times

We are looking for the ratio \(t_1 : t_2\). Divide the equations for \(t_1\) and \(t_2\), we get: \(\frac{t_1}{t_2} = \frac{a_2}{a_1}\) Using the value of \(\frac{a_1}{a_2} = \frac{5}{2}\) from Step 1, we can calculate the ratio of times: \(\frac{t_1}{t_2} = \frac{a_2}{a_1} = \frac{2}{5}\) So the ratio of times required for the two masses to reach the same final velocity is \(2:5\). The correct answer is (D) \(2: 5\).

Key concepts

Acceleration

Acceleration can be understood as the rate at which an object's velocity changes over time. It is directly influenced by both the force applied to the object and the object's mass. According to Newton's second law of motion,

• Acceleration (\( a \)) is defined by the formula \( F = ma \), where \( F \) represents the force applied to the object, and \( m \) is the object's mass.
• This means that for a constant force, the acceleration is inversely proportional to the object’s mass.

In the exercise, we calculated the acceleration of two bodies with different masses, 2 kg and 5 kg, subjected to the same force. By setting up the ratio of their accelerations, we could see how their respective masses affect their accelerations.
The formula that helped us understand this relationship was \( \frac{a_1}{a_2} = \frac{m_2}{m_1} \). This shows that a lighter object (smaller mass) will accelerate more than a heavier object if the force remains constant.
Thus, with the mass ratio in hand, determining the acceleration relationship is straightforward, as evident in our result of the ratio being 5:2.

Mass and Force Relationship

The relationship between mass and force is a crucial part of understanding motion dynamics. Newton’s second law states that the force applied to an object is the product of its mass and its acceleration. Let's break down this relationship:

• If you increase the mass of an object while applying the same force, the acceleration decreases, indicating an inversely proportional relationship.
• On the other hand, if the force changes and mass remains constant, the acceleration will alter proportionally to the change in force.

In our exercise, the same force was applied to two different masses. The law \( F = m_1a_1 = m_2a_2 \) allowed us to equate the different scenarios for each mass. This principle helps understanding why different objects may move differently under the same force. The heavier mass, being less responsive, requires more time to achieve the same velocity compared to a lighter mass. These intricate mass-force interactions underpin many mechanical and physical phenomena.

Final Velocity Calculation

Calculating final velocity involves understanding how objects in motion respond to applied forces over time. The formula for velocity

• is given as \( v = u + at \), where \( v \) is the final velocity, \( u \) is the initial velocity, and \( a \) is the acceleration applied over time \( t \).
• If the object is initially at rest, then \( u = 0 \), simplifying our formula to \( v = at \).

In the context of our problem, both objects started from rest, thus greatly simplifying calculations. By knowing the acceleration from the mass and force relationship, time required to reach a given final velocity can be calculated as \( t = \frac{v}{a} \).
This relationship allowed us to determine how much time each of the two masses required to reach the same velocity. By equating the terms for time, we deduced the accurate ratio of their time \( t_1 : t_2 \) as 2:5, indicating the heavier object requires significantly longer to reach the same velocity. Final velocity calculations, thus, become a cornerstone in kinematic equations used to predict an object's future position or speed.