Question

3. The displacement x of a particle along X-axis is
given by x = 3 + 8 + 712. Obtain its velocity and
acceleration att = 2s. (Ans. 36 ms -1, 14 ms 2)
Or​

Answer 1

GIVEN:

• Displacement, x = 3 + 8t + 7t²
• Time, t = 2 s

TO FIND:

• Velocity, v
• Acceleration, a

FORMULAE:

• v = dx/dt

• a = dv/dt

SOLUTION:

STEP 1: To find velocity

$v = \frac{dx}{dt} \\ \\ v = \frac{d}{dt} (3 + 8t + 7 {t}^{2} ) \\ \\ v = 0 + 8(1) + (7 \times 2 \times t) \\ \\ v = 8 + 14t \\ \\ v = 8 + (14 \times 2) \\ \\ v = 8 + 28 \\ \\ v = 36 \: \frac{m}{s}$

Velocity at 2 second is 36 m/s.

STEP 2: To find acceleration.

$a = \frac{dv}{dt} \\ \\ a = \frac{d}{dt} (8 + 14t) \\ \\ a = 0 + (14 \times 1) \\ \\ a = 0 + 14 \\ \\ a = 14 \: \frac{m}{ {s}^{2} }$

Acceleration at 2 seconds is 14 m/s².

ANSWER:

• Velocity at 2 second is 36 m/s.

• Acceleration at 2 seconds is 14 m/s².

POINTS TO NOTICE:

• Differentiation of displacement gives instantaneous velocity.

• Differentiation of velocity gives instantaneous acceleration.

• Integration of acceleration gives the change in velocity.

• Integration of velocity gives the displacement.

• Area under velocity time graph gives displacement.

• Slope of velocity time graph gives acceleration.

• Slope of displacement time graph gives velocity.

• Area under acceleration time graph gives velocity.