1. Projectile motion A particle P of constant mass m has position 1'03) = $05) 3' + W) 3', where 2' and j are the base vectors of a global Cartesian coordinate system in an inertial frame of reference. The particle is inuenced by a uniform gravitational eld gj . At time t = 0, the particle is at the origin of the coordinate system and is projected with speed U at an angle 0 S 9 3 7r to the base vector i. (a) Use Newton's second law of motion to show that 1 $05) = Ut cos 9, y(t) = Ut sint9 Egg. (b) i. Find the time T at which the particle returns to ground level (2/ = 0)- ii. Find the distance R of the particle from the origin at time 7'. iii. For xed U , nd the angle 9 that maximises the distance R. (c) Starting from Newton's second law, derive the energy equation 1 5W1\"? + mgy = E, and verify that the energy E is conserved during the motion