Question

A force \( \mathbf{F} = ai + bj + ck \) is acting on a body of mass \( m \). The body was initially at rest at the origin. The co-ordinates of the body after time \( t \) will be:

• A. \( \frac{ar^2}{2m} i + \frac{br^2}{2m} j + \frac{cr^2}{2m} k \)
• B. \( \frac{ar^2}{2m} i + \frac{br^2}{2m} j + \frac{cr^2}{2m} k \)
• C. \( \frac{ar}{m} i + \frac{br}{m} j + \frac{cr}{m} k \)
• D. \( \frac{ar}{m} i + \frac{br}{m} j + \frac{cr}{m} k \)

Hint:

To find the position vector in motion under a constant force, integrate the acceleration twice and apply the initial conditions such as rest at the origin.

Answer 1

The Correct Option is A

Step 1: Understanding the Force Equation The force acting on the body is given as: \[ \mathbf{F} = ai + bj + ck \] Where: - \( a, b, c \) are constants representing the components of the force in the \( x, y, \) and \( z \) directions, - \( i, j, k \) are unit vectors along the \( x, y, \) and \( z \) axes, respectively. Step 2: Newton's Second Law of Motion From Newton's second law, we know that: \[ \mathbf{F} = m \cdot \mathbf{a} \] Where: - \( \mathbf{a} \) is the acceleration vector, and - \( m \) is the mass of the body. The acceleration is the derivative of velocity with respect to time: \[ \mathbf{a} = \frac{d\mathbf{v}}{dt} \] And velocity is the derivative of displacement: \[ \mathbf{v} = \frac{d\mathbf{r}}{dt} \] Step 3: Integrating to Find Displacement To find the displacement \( \mathbf{r}(t) \), we need to integrate the acceleration twice because acceleration is the second derivative of displacement with respect to time: \[ \mathbf{r}(t) = \int \mathbf{v}(t) dt = \int \left( \int \mathbf{a}(t) dt \right) dt \] Since \( \mathbf{F} = m \cdot \mathbf{a} \), the acceleration is: \[ \mathbf{a} = \frac{F}{m} = \frac{ai + bj + ck}{m} \] The displacement is the integral of acceleration over time. The solution after two integrations (since the body starts at rest at the origin) will give the coordinates as: \[ \mathbf{r}(t) = \frac{1}{2} \cdot \left( \frac{ar^2}{m} \right) i + \frac{1}{2} \cdot \left( \frac{br^2}{m} \right) j + \frac{1}{2} \cdot \left( \frac{cr^2}{m} \right) k \] Step 4: Conclusion Therefore, the coordinates of the body after time \( t \) are given by: \[ \boxed{(A) \, \frac{ar^2}{2m} i + \frac{br^2}{2m} j + \frac{cr^2}{2m} k} \]