Question

A particle is moving along straight line with initial velocity \(+7 \mathrm{~m} / \mathrm{sec}\) and uniform acceleration \(-2 \mathrm{~m} / \mathrm{sec}^{2}\). The distance travelled by the particle in \(4^{\text {th }}\) second of its motion is : (1) zero (2) \(0.25 \mathrm{~m}\) (3) \(0.5 \mathrm{~m}\) (4) \(7 \mathrm{~m}\) (5) \(0.75 \mathrm{~m}\)

Short answer

The distance traveled in the 4th second is 0 m.

Step-by-step solution

Understand the Problem

A particle moves with an initial velocity of \(+7 \, \text{m/s}\) and has a uniform acceleration of \(-2 \, \text{m/s}^2\). We need to determine the distance traveled in the 4th second of its motion.

Identify the Formula

To solve this, use the formula for distance traveled in the nth second: \[ S_n = u + a(n - 0.5) \] where \(S_n\) is the distance traveled in the nth second, \(u\) is the initial velocity, \(a\) is the acceleration, and \(n\) is the specific second.

Substitute the Given Values

For the 4th second (n = 4), initial velocity \(u = +7 \, \text{m/s}\), and acceleration \(a = -2 \, \text{m/s}^2\), substitute the values into the formula: \[ S_4 = 7 + (-2)(4 - 0.5) \]

Solve the Equation

Calculate it step-by-step: 1. Calculate \(4 - 0.5\): \[ 4 - 0.5 = 3.5 \] 2. Multiply by acceleration \[ -2 \times 3.5 = -7 \] 3. Add the initial velocity \[ 7 + (-7) = 0 \]

Conclude the Result

The calculated distance traveled in the 4th second is \(0 \, \text{m}\). Thus, the correct answer is option (1).

Key concepts

kinematics

Kinematics is a branch of physics that deals with the motion of objects without considering the forces that cause the motion. It focuses on the position, velocity, and acceleration of objects. When studying kinematics, you typically use equations of motion to describe how objects move. These equations can help you predict where an object will be at a given time or how fast it will be moving.

uniform acceleration

Uniform acceleration refers to a constant acceleration, meaning the object's velocity changes at a constant rate over time. An important concept here is the relationship between velocity, acceleration, and time. If an object has a uniform acceleration, we can use specific equations of motion to describe its behavior. These include:

• Final velocity: \(v = u + at\)
• Distance traveled: \(d = ut + \frac{1}{2}at^2 \)
• Distance in nth second: \(S_n = u + a(n - 0.5) \)

Understanding these equations allows you to solve various problems related to uniformly accelerating objects.

initial velocity

The initial velocity (denoted as \(u\)) is the velocity at which an object starts its motion. This is a crucial parameter because it affects how the object moves over time. In kinematic problems, knowing the initial velocity is essential for predicting future position and velocity. For instance, in our problem:

• Initial velocity (u): \(u = +7 \text{m/s}\)


• It influenced the calculation of distance traveled in the fourth second.

By understanding initial velocity, you can better understand an object's motion from the start point.