Question

A particle moves along a straight line OX. At a time t (in seconds) the distance x (in metres) of the particle from O is given by x = 40 + 12t - t³ How long would the particle travel before coming to rest ?

• A. 14 m
• B. 28 m
• C. 56 m
• D. 70 m

Answer 1

The Correct Option is C

To determine how long the particle travels before coming to rest, we need to analyze the motion described by the given expression for distance:

x = 40 + 12t - t^3

The particle comes to rest when its velocity becomes zero. The velocity of the particle is the derivative of distance with respect to time:

v(t) = \frac{dx}{dt} = \frac{d}{dt}(40 + 12t - t^3)

Computing the derivative, we get:

v(t) = 12 - 3t^2

To find when the particle comes to rest, set the velocity v(t) to zero:

0 = 12 - 3t^2

Solving for t, we rearrange and solve the equation:

3t^2 = 12 \implies t^2 = 4 \implies t = \pm 2

Since time cannot be negative, we take t = 2 seconds.

Next, we plug this value back into the expression for distance to find how far the particle has traveled when it comes to rest:

x = 40 + 12(2) - (2)^3

Calculate step-by-step:

• 12 \times 2 = 24
• (2)^3 = 8
• x = 40 + 24 - 8 = 56 meters

Therefore, the particle travels 56 meters before coming to rest. The correct answer is:

• 56 m