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: Force Equation The force on an object is: \[ \mathbf{F} = ai + bj + ck \] Where: - \( a, b, c \) are constants (force components in x, y, and z directions). - \( i, j, k \) are unit vectors along the x, y, and z axes. Step 2: Newton's Second Law Newton's second law states: \[ \mathbf{F} = m \cdot \mathbf{a} \] Where: - \( \mathbf{a} \) is the acceleration vector. - \( m \) is the object's mass. Acceleration is the time derivative of velocity: \[ \mathbf{a} = \frac{d\mathbf{v}}{dt} \] Velocity is the time derivative of displacement: \[ \mathbf{v} = \frac{d\mathbf{r}}{dt} \] Step 3: Displacement Calculation To find displacement \( \mathbf{r}(t) \), integrate acceleration twice: \[ \mathbf{r}(t) = \int \mathbf{v}(t) dt = \int \left( \int \mathbf{a}(t) dt \right) dt \] Since \( \mathbf{F} = m \cdot \mathbf{a} \), acceleration is: \[ \mathbf{a} = \frac{F}{m} = \frac{ai + bj + ck}{m} \] Displacement is the integral of acceleration over time. With initial conditions (at rest at origin), the solution is: \[ \mathbf{r}(t) = \frac{1}{2} \cdot \left( \frac{ar^2}{m} \right) i + \frac{1}{2} \cdot \left( \frac{br^2}{m} \right) j + \frac{1}{2} \cdot \left( \frac{cr^2}{m} \right) k \] Step 4: Conclusion Therefore, the coordinates of the body after time \( t \) are: \[ \boxed{(A) \, \frac{ar^2}{2m} i + \frac{br^2}{2m} j + \frac{cr^2}{2m} k} \]