Question

The displacement x of a body varies with time t as x = - (2/3) t^2 + 16t +2 in what time the body comes to rest ? (x is measured in meter and t in seconds).​

Answer 1

12 seconds is the required answer.

GIVEN:

• Displacement , x = - (2/3) t² + 16t +2

TO FIND:

Time at which body comes to rest , t

FORMULA:

Differentiation of displacement gives instantaneous velocity.

v = dx/dt

SOLUTION:

• When the body comes to rest, it means there is zero velocity.

• So equating the velocity equation to zero can give the time at which the body comes to rest.

STEP 1: TO FIND VELOCITY:

$x = - \frac{2}{3} {t}^{2} + 16t + 2 \\ \\ v = \frac{dx}{dt} \\ \\ v = \dfrac{d( - \frac{2}{3} {t}^{2} + 16t + 2 )}{dt} \\ \\ v = - \frac{2}{3} (2)(t) + 16 + 0 \\ \\ v = - \frac{4}{3} t + 16$

STEP 2: TO FIND TIME:

$v = 0 \\ \\ - \frac{4}{3} t + 16 = 0 \\ \\ \frac{4}{3} t = 16 \\ \\ t = 16 \times \frac{3}{4} \\ \\ t = \frac{16}{4} \times 3 \\ \\ t = 4 \times 3 \\ \\ t = 12 \: seconds$

ANSWER:

Time at which body comes to rest is 12 seconds.

IMPORTANT POINTS:

• Differentiation of displacement gives instantaneous velocity.

• Differentitation of velocity gives instantaneous acceleration.

• Integration of acceleration gives change in velocity.

• Integration of velocity gives displacement.

• Slope of displacement time graph gives instantaneous velocity.

• Slope of velocity time graph gives instantaneous acceleration.

• Area under acceleration time graph gives change in velocity.

• Area under velocity time graph gives displacement.