Question

A particle moves along the X-axis according to the equation x=4t^2+5t+16 where x is in meters and t is in seconds. The acceleration of the particle in m/s2 will be

Answer 1

Solution :

⏭ Given:

✏ Equation of motion has been provided.

$\boxed{ \red{ \tt{x = 4 {t}^{2} + 5t + 16}}}$

• x is in meter unit.
• t is in second unit.

⏭ To Find:

✏ Acceleration of particle

⏭ Concept:

$\star \tt \: \blue{velocity(v) = \dfrac{dx}{dt}} \\ \\ \star \tt \: \green{acceleration(a) = \dfrac{dv}{dt}} \\ \\ \star \tt \: \pink{acceleration(a) = \dfrac{ {d}^{2} x}{d {t}^{2} }}$

⏭ Calculation:

$\dashrightarrow \sf \: x = 4 {t}^{2} + 5t + 16 \\ \\ \sf \purple{ \dag \: differenciating \: both \: sides \: wrt \: t} \\ \\ \dashrightarrow \sf \: v = \dfrac{dx}{dt} = \dfrac{d(4 {t}^{2} + 5t + 16 )}{dt} \\ \\ \dashrightarrow \sf \: v = 8t + 5 \\ \\ \sf \purple{ \dag \: again \: differenciating \: both \: sides \:wrt \: t } \\ \\ \dashrightarrow \sf \: a = \dfrac{dv}{dt} = \dfrac{d(8t + 5)}{dt} \\ \\ \dashrightarrow \: \boxed{ \tt{ \orange{a = 8 \: m {s}^{ - 2}}}} \: \red{ \star}$

✏ Here, acceleration has constant value throughout the whole journey.

Answer 2

Answer:

Given:

Displacement vs time Equation has been given as follows :

$x = 4 {t}^{2} + 5t + 16$

To find:

Acceleration of the particle in m/s²

Concept:

Acceleration is an vector quantity representing the rate of change of velocity with time. It has both magnitude and direction.

Acceleration can be expressed in various forms :

• in terms of velocity and displacement {v(dv/dx)}
• in terms of only Displacement (d²x/dt²)
• in terms of only velocity (dv/dt)

Calculation:

$x = 4 {t}^{2} + 5t + 16$

$= > v = \dfrac{dx}{dt}$

$= > v = \dfrac{d(4 {t}^{2} + 5t + 16)}{dt}$

Using additive property of differentiation :

$= > v = 8t + 5 + 0$

$= > v = 8t + 5$

performing 2nd order Differentiation :

$= > a = \dfrac{dv}{dt} = \dfrac{ {d}^{2} x}{d {t}^{2} }$

$= > a = \dfrac{d(8t + 5)}{dt}$

$= > a = 8 \: m {s}^{ - 2}$

So acceleration is constant and has a magnitude of 8 m/s²