The velocity function (in meters per second) is given for a particle moving along a line.v(t) = 3t − 8,    0 ≤ t ≤ 5(a) Find the displacement. m(b) Find the distance traveled by the particle during the given time interval. m

The acceleration function (in m/s2) and the initial velocity v(0) are given for a particle moving along a line.a(t) = 2t + 2,    v(0) = −15,    0 ≤ t ≤ 5(a) Find the velocity at time t.v(t) = m/s(b) Find the distance traveled during the given time interval.

EXAMPLE 6 A particle moves along a line so that its velocity at time t is v(t) = t2 − t − 20 (measured in meters per second).(a) Find the displacement of the particle during 1 ≤ t ≤ 6.(b) Find the distance traveled during this time period.SOLUTION(a) By this equation, the displacement iss(6) − s(1)  =  6v(t) dt1 =  6(t2 − t − 20) dt1 =  t33​ − t22 − 20t 61 =  −2756​ .This means that the particle moved approximately 45.83 meters to the left.(b) Note that v(t) = t2 − t − 20 = (t − 5)(t + 4) and so v(t) 0 on the interval [1, 5] and v(t) 0 on [5, 6]. Thus, from this equation, the distance traveled is6|v(t)| dt1 =  5[−v(t)] dt1 + 6v(t) dt5 =  5(−t2 + t + 20) dt1 + 6(t2 − t − 20) dt5 =  −t33​+t22​+20t 51 +  t55​+t22​+20t 65 =  9.17 .

The position of a particle is defined by the equation s = 0.13t3 + t2 - 3t + 2, where s is in metres and t is in seconds. This equation is valid for the time range t=0 to t=5 seconds. Determine the following:(a) the velocity of the particle at t=1.25 seconds, v = Answer m/s(b) the acceleration of the particle at t=1.25 seconds, a = Answer m/s2

A particle P is moving along a straight line through the fixed point O.The displacement, s metres, of P from O at time t seconds is given bys = t 3 – 27t + 55 t . 0(a) Write down the distance, in metres, of P from O when t = 0(1)(b) Find an expression, in terms of t, for the velocity, v m/s, of P at time t seconds.(2)(c) Find the value of t when P is closest to O.(2)(d) Find the distance, in metres, of P from O when P is closest to O.(1)(e) Find the distance, in metres, travelled by P in the interval 0 - t - 5(3)..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

A particle moves along a line with a velocity v(t)=t2+3t−4, measured in meters per second. Find the total distance the particle travels from t=0 seconds to t=2 seconds.Enter an exact answer.Provide your answer below: