Section 7-3: Collisions & Impulse

2. Section 7-3: Collisions & Impulse During a collision, objects can be deformed due to the large forces involved. • Brieflyconsider the details of a collision Assume that a collision lasts a very small time t • During the collision, the net force on the object is Newton’s 2nd Law: ∑F = p/tor p = (∑F)/t (momentum change of the object due to the collision) Define: p  Impulse Jthat the collision gives the object(change in momentum for the object!) • In the usual case: Either only one force is acting or we replace the left side by an average collision force:Fc = ∑F Impulse:J = p = Fc t Example: A tennis ball is hit by a racket as in the figure.

3. ∑Fis time dependent & the collision time tis often very small. So, not much error is made in replacing ∑Fwith an average force, Fc or Favg. The true Impulse in the collision is the area under the F vs. t curve (green in the left figure).In the approximation that Fc ≈∑F, the Impulseis approximately: J = p ≈Fct This approximation is the same as replacing the green area in the left figure by the green rectangle in the right figure. Replace the green area at the left with the green rectangle at the right.

4. The true impulse is the area under the Fc vs. t curve. t is usually very small & Fc is time dependent

5. It is often a good approximation to replace the area under the Fc vs. t curve with the area of the green rectangle. The approximate impulse is then J = p ≈Fct

6. Example 7-6 • Advantage of bending knees when landing! a)m =70 kg, h =3.0 m Impulse: p = ? Ft= p = m(0-v) First, find v (just before hitting): KE + PE = 0 m(v2 -0) + mg(0 - h) = 0  v = 7.7 m/s Impulse: p = -540 N s Just before he hits the ground  Just after he hits the ground   Opposite the person’s momentum

7. Advantage of bending knees when landing! Impulse: p = -540 N s m =70 kg, h =3.0 m, F = ? b) Stiff legged: v = 7.7 m/s to v = 0 in d = 1 cm (0.01m)! v = (½ )(7.7 +0) = 3.8 m/s Time t = d/v = 2.6  10-3 s F = |p/t| = 2.1  105 N (Net force upward on person) From free body diagram, F = Fgrd - mg  2.1  105 N Enough to fracture leg bone!!!

8. Advantage of bending knees when landing! Impulse: p = -540 N s m =70 kg, h =3.0 m, F = ? c) Knees bent: v = 7.7 m/s to v = 0 in d = 50 cm (0.5m) v = (½ )(7.7 +0) = 3.8 m/s Time t = d/v = 0.13s F = |p/t| = 4.2  103 N (Net force upward on person) From free body diagram, F = Fgrd - mg  4.9  103 N Leg bone does not break!!!

9. Example: Crash Test • Crash test: Car, m = 1500 kg, hits wall. 1 dimensional collision. +x is to the right. Before crash,v = -15 m/s. After crash, v = 2.6 m/s.Collision lasts Δt = 0.15 s. Find:Impulse car receives & average force on car. Assume: Force exerted by wall is large compared to other forces Gravity & normal forces are perpendicular & don’t effect the horizontal momentum  Use impulse approximation p1 = mv1 = -2.25 kg m/s, p2 = mv2 = 2.64 kg m/s J = Δp = p2 – p1 = 2.64  104 kg m/s (∑F)avg = (Δp/Δt) = 1.76  105 N

10. Example: Karate blow Estimate the impulse & the average force delivered by a karate blow that breaks a board a few cm thick. Assume the hand moves at roughly 10 m/s when it hits the board.

11. Problem 17 • Impulse: p = change in momentum  wall. • Momentum || to wall does not change.  Impulse will be  wall. Take + direction toward wall, Impulse = p = mv = m[(- v sinθ) - (v sinθ)] = -2mv sinθ = - 2.1 N s. Impulse on wall is in opposite direction: 2.1 N s. vi = v sinθ vf = - v sinθ