Problem 386 - JEE Physics

Question

A body of mass \(5 \mathrm{~kg}\) starts motion form the origin with an initial velocity \(\mathrm{v}_{0} \rightarrow=30 \mathrm{i}+40 \mathrm{j} \mathrm{m} / \mathrm{s}\) If a constant force \(\mathrm{F}=-\left(\mathrm{i}^{\wedge}+5 \mathrm{j}\right) \mathrm{N}\) acts on the body, than the time in which the Y-component of the velocity becomes zero is (A) \(5 \mathrm{~s}\) (B) \(20 \mathrm{~s}\) (C) \(40 \mathrm{~s}\) (D) \(80 \mathrm{~s}\)

Step-by-Step Solution

Answer

The time in which the Y-component of the velocity becomes zero is \(t = 40 \mathrm{s}\). The correct option is (C).

Step 1: Calculate the acceleration

To calculate the acceleration, we can use Newton's second law of motion, which states that Force = Mass * Acceleration, i.e., F = m*a Given that F = -i^ + 5j N and mass m = 5 kg. So, let's calculate the acceleration (a): a = F / m = (-i^ + 5j) / 5 a = -0.2i^ + 1j ms^-2 Now we have the acceleration components as a_x = -0.2 ms^-2 and a_y = 1 ms^-2.

Step 2: Find the time when the Y-component of the velocity becomes zero

To find the time when the Y-component of the velocity becomes zero, we can set the final Y-component of the velocity equal to zero, and then use the following equation of motion: v = v_0 + at We solve for t and use the given initial Y-component velocity (40 m/s) and the Y-component acceleration (1 ms^-2): 0 = 40 + (1*t) t = -40 s However, time cannot be negative, so there must be a mistake in our calculations.

Step 3: Recalculate acceleration and time

Let's re-check the given force component F_y which should be -5j instead of 5j. The correct constant force is given as F = -i^ - 5j N Now, let's calculate the correct acceleration (a): a = F / m = (-i^ - 5j) / 5 a = -0.2i^ - 1j ms^-2 Now, we have the correct acceleration components as a_x = -0.2 ms^-2 and a_y = -1 ms^-2. Now, let's find the time when the Y-component of the velocity becomes zero using the correct Y-component acceleration (-1 ms^-2): 0 = 40 + (-1*t) t = 40 s

Step 4: Check the correct option

Now we have the time when the Y-component of velocity becomes zero as 40 seconds. Comparing this to the given options: (A) 5 s (B) 20 s (C) 40 s (D) 80 s The correct answer is (C) 40 s.

Key Concepts

Newton's Second Law of Motion

Newton's Second Law of Motion is a fundamental principle in physics that helps us understand how forces affect the motion of an object. The law is concisely given by the formula:

• Force = Mass × Acceleration

This equation tells us that the force acting on an object is equal to the mass of the object multiplied by its acceleration. In simpler terms, if you apply a larger force, the object will accelerate more, and the more massive the object is, the harder it is to accelerate it.
In the given exercise, we apply this law to calculate the acceleration of a 5 kg object under a given force of \((-i^ - 5j)\) Newtons. Using the law, we find that the object's acceleration in the x-direction is \(-0.2 \, \text{ms}^{-2}\) and in the y-direction is \(-1 \, \text{ms}^{-2}\). Understanding this law is crucial for calculating the following steps in solving projectile motion problems.

Acceleration Calculation

Acceleration is a measure of how quickly an object's velocity changes. In this exercise, we need to find the acceleration components due to the given force. Given that:

• Force, \(F = -i^ - 5j \, \text{N}\)
• Mass, \(m = 5 \, \text{kg}\)

We use the formula for acceleration:

• Acceleration, \(a = \frac{F}{m}\)

Plugging in the values, we calculate:

• \(a_x = \frac{-1}{5} = -0.2 \, \text{ms}^{-2}\)
• \(a_y = \frac{-5}{5} = -1 \, \text{ms}^{-2}\)

Thus, we have the acceleration components, \(a_x\) and \(a_y\), which show how the force affects the object in each direction. This step is pivotal, as it provides the necessary values to determine how the velocity components will change over time.

Components of Velocity

Velocity is a vector quantity, meaning it has both magnitude and direction. In this exercise, you start with an initial velocity \(\mathbf{v}_{0} = 30\mathbf{i} + 40\mathbf{j} \, \text{m/s}\). This indicates that initially, the object is moving with:

• An x-component of velocity of 30 m/s
• A y-component of velocity of 40 m/s

To study how an external force modifies these components, we separate these directions and work on them independently. The change in each component is influenced by the corresponding directional acceleration.
For instance, due to the acceleration \(-1 \, \text{ms}^{-2}\) in the y-direction, the y-component of velocity will reduce until it becomes zero. Understanding these components and their changes helps predict the trajectory of the object under constant force.

Equation of Motion

The equation of motion is key in predicting how an object’s velocity is altered over time under constant acceleration. In this case, the equation \(v = v_0 + at\) helps find when the y-component of the velocity becomes zero.
We are given:

• Initial y-velocity, \(v_{0y} = 40 \, \text{ms}^{-1}\)
• y-velocity to be zero (\(v_y = 0\))
• y-acceleration, \(a_y = -1 \, \text{ms}^{-2}\)

By substituting the values into the equation: \0 = 40 + (-1)twe solve:

• t = 40 \, \text{seconds}

This calculation provides that the y-component velocity zeroes after 40 seconds under the provided conditions. The equation of motion is thus an essential tool in problems involving changes in velocity, allowing us to precisely calculate time intervals for velocity changes.