To solve the problem of finding the displacement of a particle whose velocity [tex]\( v \)[/tex] varies with time [tex]\( t \)[/tex] as [tex]\( v = t^2 \)[/tex] from [tex]\( t = 0 \)[/tex] to [tex]\( t = 2 \)[/tex] seconds, we need to integrate the velocity function over the given time interval. Here's the step-by-step solution:
1. Given Information:
- Velocity function: [tex]\( v(t) = t^2 \)[/tex]
- Time interval: [tex]\( t \)[/tex] ranges from 0 to 2 seconds.
2. Concept:
- Displacement is obtained by integrating the velocity function with respect to time over the given interval.
3. Mathematical Formulation:
- Displacement [tex]\( s \)[/tex] is given by the integral of [tex]\( v(t) \)[/tex] from [tex]\( t = 0 \)[/tex] to [tex]\( t = 2 \)[/tex]:
[tex]\[ s = \int_{0}^{2} v(t) \, dt \][/tex]
[tex]\[ s = \int_{0}^{2} t^2 \, dt \][/tex]
4. Integrate the Function:
- To find the integral, we use the integral of [tex]\( t^2 \)[/tex]:
[tex]\[ \int t^2 \, dt = \frac{t^3}{3} + C \][/tex]
where [tex]\( C \)[/tex] is the constant of integration, which is not needed for definite integrals.
5. Evaluate the Definite Integral:
- Substitute the upper and lower limits into the integrated function:
[tex]\[ s = \left[ \frac{t^3}{3} \right]_{0}^{2} \][/tex]
[tex]\[ s = \left( \frac{2^3}{3} \right) - \left( \frac{0^3}{3} \right) \][/tex]
[tex]\[ s = \frac{8}{3} - 0 \][/tex]
[tex]\[ s = \frac{8}{3} \][/tex]
6. Final Result:
- The displacement [tex]\( s \)[/tex] from [tex]\( t = 0 \)[/tex] to [tex]\( t = 2 \)[/tex] seconds is [tex]\( \frac{8}{3} \)[/tex] meters, which is approximately [tex]\( 2.66666666666667 \)[/tex] meters.
Therefore, the correct answer from the given options is not listed correctly in plain fractions. Correct interpretation of the options would be:
(1) [tex]\( \frac{1}{4} \)[/tex] m is equal to 0.25 m.
(2) [tex]\( \div \)[/tex] m (this seems like a typographical error, hence dismissed).
(3) [tex]\( \frac{10}{1} \)[/tex] m is equal to 10 m.
(4) [tex]\( \frac{12}{2} \)[/tex] m is equal to 6 m.
None of these options matches our calculated answer which is [tex]\( 2.66666666666667 \)[/tex] m or [tex]\( \frac{8}{3} \)[/tex] m. Therefore, none of the answers (1), (2), (3), and (4) provided in the options match the correct displacement calculated.
