Question

A body of mass \(5 \mathrm{~kg}\) starts from the origin with an initial velocity \(u^{\rightarrow}=30 \mathrm{i}+40 \mathrm{j} \mathrm{ms}^{-1}\). If a constant Force $\underline{F}=-\left(\mathrm{i}^{\wedge}+5 \mathrm{j}\right) \mathrm{N}$ acts on the body, 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 \(40 \mathrm{s}\), which corresponds to option (C).

Step-by-step solution

Understand the terms and notations used

We are given initial velocity \(\vec{u}\) represented in vector form as \(30 \mathrm{i} + 40 \mathrm{j} \mathrm{ms}^{-1}\), where 30 and 40 are the components in the x and y directions respectively. Similarly, the constant force \(\vec{F}\) is given in vector form as \(-\mathrm{i} + 5 \mathrm{j} \mathrm{N}\), where -1 and 5 are the components in the x and y directions respectively.

Identify the equation of motion for the y-component of velocity

Since we are focusing on the y-component of the body's velocity, we know that Newton's second law states that: \(F_{y}=m a_{y}\) Where \(F_{y}\) is the y-component of force, m is the mass of the body, and \(a_{y}\) is the y-component of acceleration.

Calculate the y-component of force and acceleration

We know the y-component of force \(F_{y}\) is 5 N and the mass of the body m is 5 kg. Using this information, we can calculate the y-component of acceleration \(a_{y}\): \(a_{y}=F_{y} / m\) \(a_{y}=5 / 5\) \(a_{y}=1 \mathrm{ms}^{-2}\)

Calculate the time when y-component of velocity becomes zero

Now, we can use the equation of motion to find the time t when the y-component of velocity becomes zero. The equation of motion is: \(v_{y}=u_{y}+a_{y} t\) Since we want to find the time when the y-component of velocity \(v_{y}\) becomes zero, we set \(v_{y}=0\) and the initial y-component of velocity \(u_{y}=40 \mathrm{ms}^{-1}\). We have already calculated the y-component of acceleration \(a_{y}=1 \mathrm{ms}^{-2}\). Plugging these values into the equation gives: \(0 = 40 + 1*t\)

Solve for time t

From the previous step, we have the equation: \(0 = 40 + 1*t\) Solving for t: \(-40 = 1*t\) \(t=-40\) However, we cannot have a negative time value. The negative sign indicates that the force component is in the opposite direction, causing the velocity to decrease and reach zero. Therefore, the correct answer is: \(t=|(-40)| = 40 \mathrm{s}\) The time in which the y-component of the velocity becomes zero is 40 s, which corresponds to option (C).