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) \(80 \mathrm{~s}\)

Short answer

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

Step-by-step solution

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.

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.

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

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.