Question

The distance travelled by a particle is related to time $t$ as $x=4 t^2$ The velocity of the particle at $t-5 s$ is:-

• A. $8 \,m s^{-1}$
• B. $20\, ms ^{-1}$
• C. $25\, ms ^{-1}$
• D. $40\, ms ^{-1}$

Answer 1

The Correct Option is D

To find the velocity of the particle at time \( t = 5 \, \text{s} \), we need to first determine the expression for velocity in terms of time \( t \). Given the equation for the distance traveled by the particle:

\(x = 4t^2\)

we know that velocity \( v \) is the derivative of distance with respect to time, that is:

\(v = \frac{dx}{dt}\)

Calculate the derivative of \( x \) with respect to \( t \):

\(\frac{dx}{dt} = \frac{d}{dt}(4t^2) = 8t\)

This shows that the velocity \( v \) as a function of time \( t \) is:

\(v = 8t\)

Now, substitute \( t = 5 \, \text{s} \) into the velocity equation:

\(v = 8 \times 5 = 40 \, \text{m/s}\)

Therefore, the velocity of the particle at \( t = 5 \, \text{s} \) is \( 40 \, \text{m/s} \).

Hence, the correct answer is $40 \, \text{m/s}$.

To disqualify the other options:

• \(8 \, \text{m/s}\): This would correspond to a lower \( t \), such as \( 1 \, \text{s} \), not \( 5 \, \text{s} \).
• \(20 \, \text{m/s}\): This velocity would occur at \( t = 2.5 \, \text{s} \).
• \(25 \, \text{m/s}\): This velocity would occur at \( t = 3.125 \, \text{s} \).

To summarize, the correct velocity corresponding to the time \( t = 5 \, \text{s} \) is indeed \( 40 \, \text{m/s} \).