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).

Key concepts

Vector Components

When working with vectors in physics, especially those involving forces and velocities, understanding vector components is crucial. Vectors are quantities defined by both magnitude and direction. For instance, a velocity vector like \( \vec{u} = 30 \mathrm{i} + 40 \mathrm{j} \mathrm{ms}^{-1} \) indicates movement along the x-axis and y-axis separately.
This breaks down the vector into its components:

• The x-component: \( 30 \mathrm{i} \), representing velocity in the horizontal direction.
• The y-component: \( 40 \mathrm{j} \), representing velocity in the vertical direction.

This decomposition aids in independently analyzing motion or force in each direction, simplifying complex problems.

Equations of Motion

Equations of motion are mathematical expressions that describe the motion of an object. In this exercise, the focus is on the equation addressing the velocity's y-component. As per Newton's laws, the relation is given by:
\( v_{y} = u_{y} + a_{y} t \)Here,

• \(v_{y}\) represents the final velocity in the y-direction.
• \(u_{y}\) is the initial velocity in the y-direction, given as \(40 \mathrm{ms}^{-1}\).
• \(a_{y}\) is the y-component of acceleration.
• \(t\) is the time.

Using this equation, you can determine the time when the velocity component becomes zero by setting \(v_{y} = 0\) and solving for \(t\). This practical application shows how equations of motion help predict future states of objects under certain forces.

Force and Acceleration

Newton's Second Law succinctly relates force to acceleration. In vector terms, it is expressed as:
\( \vec{F} = m \vec{a} \)Force \( \vec{F} \) and mass \( m \) are vectors and scalars respectively, leading to acceleration \( \vec{a} \), also a vector. Understanding this relationship is essential to solving motion problems, where adding force results in corresponding acceleration. For only the y-component:
\( F_{y} = m a_{y} \)By solving for \(a_{y}\), you isolate the effect of force on just the vertical direction, helping us compute changes in velocity over time. The acceleration we calculated was \(1 \mathrm{ms}^{-2}\), causing the object's velocity to slow down until it reverses the motion direction or comes to a stop.

Velocity

In physics, velocity is not simply speed; it encompasses both magnitude and direction. The vector \( \vec{u} = 30 \mathrm{i} + 40 \mathrm{j} \mathrm{ms}^{-1} \) indicates initial velocities in the x and y directions. Over time, forces acting upon an object change velocities.
Considering our exercise, the initial y-velocity \(u_{y}\) was \(40 \mathrm{ms}^{-1}\), and under the influence of a force, it changes. The acceleration impacts this unless it becomes zero or surpasses it by acting in the opposite direction. That's the crux of understanding how forces and velocities interplay in motion prediction. Velocity vectors guide us through examining how objects will travel over time based on external impacts like forces and resultant accelerations.