A particle with mass m 15 is moving along a straight trajec

A particle with mass m = 1.5 is moving along a straight trajectory. The distance covered by a particle is described by the function S(t) = t^3 + t, where t > 0 is the time. (a) Kinetic energy E of a particle is given by E = mv^2/2, where v is the particle speed. Find a function that describes the kinetic energy of this particle. (b) Find kinetic energy of the particle at time t = 10. (c) What is the distance covered by the particle by time t = 7? (d) Find approximate value of time t when the speed of the particle is 25. (e) Find a function that describes the acceleration of this particle. (f) Find the acceleration of the particle at t = 6. (g) Find the time when the acceleration of the particle is 18.

Solution

m= 1.5 , S(t) = t³ + t , straight trajectory

V(t) = dS/dt = 3t²+1 ,

(a) K.E (t) = (1/2)mV(t)² = (1/2)(1.5) (3t²+1)² = 0.75(3t²+1)²

(b) K.E(10) = 0.75 (301)² = 67950.75 units

(c) S(7) = 7³ + 7 = 350 units

(d) V(t) = 25 , (3t²+1) = 25 => t² = 8 => t = 22 units

(e) a(t) = dV/dt = 6t , a(t) = 6t

(f) a(6) = 6(6) = 36 units

(g) a(t) = 18 , 6t = 18 => t = 3 units