Question

the motion of a body is given by x= t^3+4t^2-2t+5 where x is in metre find (a) velocity and acceleration at t=4s (b) average velocity and average acceleration during t =0s to t=4s

Answer 1

(a) velocity is 78m/s.
acceleration is
32m/s^2

Answer 2

Answer:

${\pink{\green{\sf{VELOCITY=78m/s}}}}$

${\pink{\green{\sf{ACCELERATION=32m/s^{2}}}}}$

${\pink{\green{\sf{AVG\:VELOCITY=30m/s}}}}$

${\pink{\green{\sf{AVG\:ACCELERATION=20m/s^{2}}}}}$

Explanation:

$\huge{\pink{\green{\underline{\red{\sf{SOLUTION-}}}}}}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: { \orange{given}} \\ { \green{x = {t}^{3} + 4 {t}^{2} - 2t + 5(position \: of \: a \: particle }} \\ \\ { \blue{to \: find}} \\ { \red{acceleration =? }} \\ { \red{velocity =? }} \\ { \red{average \: acceleration = ?}} \\ { \red{average \:velocity = ?}}$

▪According to the given question:

▪for finding velocity if position of a particle is given so we differentiate position of a particle.

$\to x = {t}^{3} + 4 {t}^{2} -2t+5\\ \to \frac{dv}{dt} = 3 {t}^{2} + 8t - 2 \\ \\ \to at \: (t = 4s) = 3 \times {4}^{2} + 8 \times 4 - 2 \\ \to velocity = 3 \times 16 + 32 - 2 \\ { \green{\therefore velocity = 78m/s}}$

▪for finding acceleration if we get velocity of a particle so again differentiate the velocity of particle.

$v=3{t}^{2}+8t-2\\\to acceleration = \frac{da}{dt} = 6t+8 \\ \to {acc}^{n} = 6t + 8 \\ \\ \to at \: (t = 4s) = 6 \times 4 + 8 \\ { \green{\therefore {acc}^{n} = 32 m/s^{2} }}$

▪Avg velocity during t=0 to t=4s.

$\to initial \: displacement(at \: t = 0) = {0}^{3} + 4 \times {0}^{2} - 2 \times 0 + 5 \\ \to u = 5m \\ \\ \to final \: displacement (at \: t = 4) = {4}^{3} + 4 \times {4}^{2} - 2 \times 4 + 5 \\ \to v = 125m \\ \\ \to avg \: velocity = \frac{v - u}{t2 - t1} \\ \to avg \: velocity = \frac{125 - 5}{4 - 0} \\ { \green{\therefore avg \: velocity = 30 m/s }}$

▪Avg acceleration during t=0 to t=4s

$\to avg \: acceleration = \frac{ v_{f} - v_{i} }{ t_{f} - t_{i} } \\ \to avg \: acceleration = \frac{78 - ( - 2)}{4 - 0} \\ { \green{\therefore avg \: acceleration = 20 m/{s}^{2} }}$

_________________________________________