Question

If the position of the particle is \( \mathbf{r} = 3 \hat{i} + 2 t^2 \hat{j} \), then the magnitude of its velocity at \( t = 5 \) second in \( {ms}^{-1} \) is:

• A. 20
• B. 10
• C. 40
• D. 50
• E. 30

Hint:

The velocity is the derivative of the position vector with respect to time. For each component, differentiate and find the magnitude of the velocity vector.

Answer 1

The Correct Option is A

The position vector of the particle is given by: \[ \mathbf{r}(t) = 3 \hat{i} + 2 t^2 \hat{j} \] To find the velocity, we differentiate the position vector with respect to time: \[ \mathbf{v}(t) = \frac{d}{dt} \mathbf{r}(t) \] Differentiating each component: \[ \mathbf{v}(t) = \frac{d}{dt} (3 \hat{i}) + \frac{d}{dt} (2 t^2 \hat{j}) \] \[ \mathbf{v}(t) = 0 \hat{i} + 4 t \hat{j} \] Thus, the velocity vector is: \[ \mathbf{v}(t) = 4 t \hat{j} \] At \( t = 5 \) seconds: \[ \mathbf{v}(5) = 4 \times 5 \hat{j} = 20 \hat{j} \, {m/s} \] The magnitude of the velocity is: \[ |\mathbf{v}(5)| = 20 \, {m/s} \] Thus, the magnitude of the velocity at \( t = 5 \) seconds is \( 20 \, {ms}^{-1} \). \[ \boxed{20} \]