Principle of Work and Energy T1 + SU1-2 = T2 Conservation of Energy

CHAPTER 15 Kinetics of a Particle: Impulse and Momentum

Objectives To develop the principle of linear impulse and momentum for a particle 2. To study the conservation of linear momentum for a particle 3. To analyze the mechanics of impact

v = ds/dt a = dv/dt dt = ds/v dt = dv/a ads = vdv T1 + SU1-2 = T2 Principle of Work and Energy

ads = vdv a = dv/dt Principle of Linear Impulse and Momentum

Linear Impulse Variable Force Constant Force

+ =

Principle of Linear Impulse and Momentum a particle system of particles

ex1 100 kg stone is originally at rest on the smooth horizontal space. If a towing force of 200 N, acting at an angle of 45o, is applied to the stone for 10 s, determine the final velocity and the normal force which the surface exerts on the stone during the time interval.

Principle of Impulse and momentum +

Principle of Impulse and momentum +

The 250 N crate is acted upon by a force having a variable magnitude P = 100t, where t is in seconds. Determine the crate’s velocity 2 s after P has been applied. The initial velocity is v1 = 1 m/s down the plane, and mk = 0.3.

Principle of Linear Impulse and Momentum a particle system of particles

Conservation of Linear Momentum for a System of Particles When the sum of the external impulses acting on a system of a particle is zero, Nonimpulsive force : - impulse that may be neglected - considered equal to zero Impulsive : - forces that are very large and act for a very short period of time produce significant change in momentum

The force of the rackets on the ball is impulsive The ball’s weight will have the negligible effect,  nonimpulsive

Ex1 Boxcar A : 15 Mg Boxcar B : 12 Mg If the car meets and couple together, determine (a) the speed of both cars just after the coupling (b) the average force between them if the coupling takes place in 0.8 s

(a) the speed of both cars just after the coupling Conservation of Linear Momentum

(b) the average force between them if the coupling takes place in 0 (b) the average force between them if the coupling takes place in 0.8 s Principles of Impulse and Momentum

Ex 2 The 600 kg cannon shown fires a 4 kg projectile with a muzzle velocity 450 m/s relative to the ground. If firing takes place in 0.03 s, determine (a) the recoil velocity of the cannon just after firing (b) the average impulsive force acting on the projectile The cannon support is fixed to the ground, and the horizontal recoil of the cannon is absorbed by two springs.

Mc = 600 [kg] Mp = 4 [kg] (Vp )1 = 450 [m/s] t = 0.03 [s] Conservation of Linear Momentum

Priciples of Impulse and Momentum

Lab test 2 : 12/04/07 (Khamis) Mid-Term 2 : 16/04/07 (Isnin)

Objectives To develop the principle of linear impulse and momentum for a particle 2. To study the conservation of linear momentum for a particle 3. To analyze the mechanics of impact

Impact Impact occurs when two bodies collide with each other during a very short period of time, causing relatively large (impulsive ) forces to be exerted between the bodies.

Central Impact

Coefficient of Restitution e e = 1 : Perfectly elastic impact e = 0 : Plastic impact / inelastic impact

Ex The bag A, having a mass of 6 kg, is released from rest at the position q = 0o. After falling to q = 90o, it strikes an 18 kg box B. If the coefficient of restitution between the bag and box is e = 0.5, determine (i) the velocities of the bag and box just after impact (ii) the loss of energy during collision

the velocities of the bag and box just after impact Conservation of Energy 6 kg Datum at q = 0o

Conservation of Momentum 6 kg Conservation of Momentum (i)

(vA)2 = - 0.554 m/s = 0.554 m/s  (vB)2 = 1.66 m/s  Coefficient of restitution (ii) Solving (i) and (ii), (vA)2 = - 0.554 m/s = 0.554 m/s  (vB)2 = 1.66 m/s 

(ii) the loss of energy during collision

(vA)2, Oblique Impact The particles move away with velocities having unknown directions and unknown magnitudes. (vA)2, (vB)2 q2 f2 If v1 velocities are known;

Ex The smooth disks A and B, having a mass of 1 kg and 2 kg, respectively, collide with the velocities shown in the figure. If e = 0.75, determine the x and y components of the final velocity of each disk just after collision.

x components y components

(i) for x components of the final velocity; Conservation of “x” Momentum (i)

Coefficient of Restitution (x) Both disk are assumed to have components of velocity in the +x direction after collision (ii)

(i) for x components of the final velocity; Solving (i) and (ii), Ans. Ans.

(ii) for y components of the final velocity; Conservation of “y” Momentum

A man kicks the 200g ball such that it leaves the ground at an angle of 30o With the horizontal and strikes the ground at the same elevation a distance of 15m away. Determine the impulse of hiss foot F on the ball. Neglect the Impulse caused by the ball’s weight while its being kicked.