Question

A 4 kg mass moves under the influence of a force \(\vec{F} = (4t^3\hat{i} - 3t^2\hat{j})\) N where t is the time in second. If mass starts from origin at t=0, the velocity and position after t = 2 s will be:

• A. \(\vec{v} = 4\hat{i} - \frac{3}{2}\hat{j}\), \(\vec{r} = \frac{8}{5}\hat{i} - \hat{j}\)
• B. \(\vec{v} = 4\hat{i} - 2\hat{j}\), \(\vec{r} = \frac{8}{5}\hat{i} - \hat{j}\)
• C. \(\vec{v} = 4\hat{i} - 3\hat{j}\), \(\vec{r} = \frac{8}{5}\hat{i} - 2\hat{j}\)
• D. \(\vec{v} = 4\hat{i} - 3\hat{j}\), \(\vec{r} = \frac{8}{5}\hat{i} - \hat{j}\)

Hint:

For problems involving time-dependent force, the process is always F \(\rightarrow\) a \(\rightarrow\) v \(\rightarrow\) r via division by mass and then two successive integrations. Always pay close attention to the initial conditions (\(v_0\) and \(r_0\)) as they determine the constants of integration.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given a time-dependent force acting on a mass. We need to find the velocity and position vectors at a specific time (t=2s), given the initial conditions (starts from rest at the origin).
Step 2: Key Formula or Approach:
We use the fundamental kinematic relations, integrating from acceleration: 1. Find acceleration: \(\vec{a} = \frac{\vec{F}}{m}\)
2. Find velocity by integrating acceleration: \(\vec{v}(t) = \int \vec{a}(t) dt + \vec{v}_0\)
3. Find position by integrating velocity: \(\vec{r}(t) = \int \vec{v}(t) dt + \vec{r}_0\)
The initial conditions are \(\vec{v}_0 = 0\) and \(\vec{r}_0 = 0\).
Step 3: Detailed Explanation:
1. Calculate Acceleration:
Given \(m = 4\) kg and \(\vec{F} = (4t^3\hat{i} - 3t^2\hat{j})\) N.
\[ \vec{a}(t) = \frac{\vec{F}}{m} = \frac{1}{4}(4t^3\hat{i} - 3t^2\hat{j}) = (t^3\hat{i} - \frac{3}{4}t^2\hat{j}) \, \text{m/s}^2 \] 2. Calculate Velocity:
Integrate \(\vec{a}(t)\) with respect to time from 0 to t. \[ \vec{v}(t) = \int_0^t (t'^3\hat{i} - \frac{3}{4}t'^2\hat{j}) dt' \] \[ \vec{v}(t) = \left[ \frac{t'^4}{4}\hat{i} - \frac{3}{4}\frac{t'^3}{3}\hat{j} \right]_0^t = \left( \frac{t^4}{4}\hat{i} - \frac{t^3}{4}\hat{j} \right) \, \text{m/s} \] At t = 2 s:
\[ \vec{v}(2) = \frac{2^4}{4}\hat{i} - \frac{2^3}{4}\hat{j} = \frac{16}{4}\hat{i} - \frac{8}{4}\hat{j} = (4\hat{i} - 2\hat{j}) \, \text{m/s} \] 3. Calculate Position:
Integrate \(\vec{v}(t)\) with respect to time from 0 to t. \[ \vec{r}(t) = \int_0^t \left( \frac{t'^4}{4}\hat{i} - \frac{t'^3}{4}\hat{j} \right) dt' \] \[ \vec{r}(t) = \left[ \frac{1}{4}\frac{t'^5}{5}\hat{i} - \frac{1}{4}\frac{t'^4}{4}\hat{j} \right]_0^t = \left( \frac{t^5}{20}\hat{i} - \frac{t^4}{16}\hat{j} \right) \, \text{m} \] At t = 2 s:
\[ \vec{r}(2) = \frac{2^5}{20}\hat{i} - \frac{2^4}{16}\hat{j} = \frac{32}{20}\hat{i} - \frac{16}{16}\hat{j} = \left( \frac{8}{5}\hat{i} - \hat{j} \right) \, \text{m} \] Step 4: Final Answer:
At t = 2 s, the velocity is \(\vec{v} = 4\hat{i} - 2\hat{j}\) m/s and the position is \(\vec{r} = \frac{8}{5}\hat{i} - \hat{j}\) m.
This result matches the values in option (B). Based on correct calculation, (B) is the intended answer.