Problem 10 - A Complete Resource Book in Physics for JEE Main

Question

A body of mass \(1.5 \mathrm{~kg}\) is thrown vertically upwards with an initial velocity of \(40 \mathrm{~m} / \mathrm{s}\) reaches its highest point after \(3 \mathrm{~s}\). The air resistance acting on the body during the ascent is (assuming air resistance to be uniform, \(g=10 \mathrm{~m} / \mathrm{s}^{2}\) ) (A) \(35 \mathrm{~N}\) (B) \(25 \mathrm{~N}\) (C) \(15 \mathrm{~N}\) (D) \(5 \mathrm{~N}\)

Step-by-Step Solution

Answer

The air resistance acting on the body during the ascent is \(25 \mathrm{~N}\).

Step 1: Calculate the gravitational force acting on the body

First, we need to find the gravitational force acting on the body. The gravitational force (F_g) can be calculated using the formula: \(F_g = m \times g\) where \(m\) is the mass of the body (1.5 kg) and \(g\) is the acceleration due to gravity (10 m/s^2).

Step 2: Calculate the final velocity at the highest point

Next, we need to determine the final velocity (v) of the body when it reaches its highest point. At the highest point, the body momentarily comes to rest before descending. Therefore, the final velocity at the highest point is 0 m/s.

Step 3: Calculate the acceleration due to air resistance

Now, we need to find the acceleration (a) due to air resistance acting on the body during its ascent. We can use the equation of motion: \(v = u + (a \times t)\), where \(v\) is the final velocity (0 m/s), \(u\) is the initial velocity (40 m/s), \(a\) is the acceleration due to air resistance, and \(t\) is the time taken to reach the highest point (3 s). Upon rearranging the equation, we get: \(a = \frac{v-u}{t}\)

Step 4: Calculate the acceleration due to air resistance

By plugging in the values, we can calculate the acceleration due to air resistance. We have \(v = 0 \mathrm{~m/s}\), \(u = 40 \mathrm{~m/s}\), and \(t = 3\mathrm{~s}\): \(a = \frac{0 - 40}{3}\)

Step 5: Calculate the net force acting on the body

Now, we will calculate the net force (F_net) acting on the body due to both gravity and air resistance. The net force can be determined as: \(F_\text{net} = m \times a_\text{net}\), where \(m\) is the mass of the body (1.5 kg) and \(a_\text{net}\) is the net acceleration due to both gravity and air resistance. Since the acceleration due to gravity is acting downwards and the acceleration due to air resistance is acting upwards (opposite direction), the net acceleration (\(a_\text{net}\)) will be the difference between these two accelerations. Therefore: \(a_\text{net} = g - a\)

Step 6: Calculate the net force acting on the body

By substituting the values, we can calculate the net force acting on the body. We have \(m = 1.5 \mathrm{~kg}\), \(g = 10 \mathrm{~m/s^2}\), and the calculated \(a\): \(F_\text{net} = 1.5 \times (10 - a)\)

Step 7: Calculate the air resistance force

Finally, we can calculate the air resistance force (F_a) acting on the body by using the following relation between the gravitational force, air resistance force, and the net force: \(F_\text{net} = F_g - F_a\) Upon rearranging: \(F_a = F_g - F_\text{net}\) By plugging in the values of \(F_g\) and \(F_\text{net}\) that we calculated in previous steps, we can find the air resistance force (F_a). Comparing this value with the given options, we can determine the correct answer.