Answer:
The velocity of A before impact = 17.90 m/s
Explanation:
Coefficient of restitution = (speed of seperation)/(speed of approach)
= (v₁ - v₂)/(u₂ - u₁)
where v₁ = velocity of the car A after the impact = ?
v₂ = velocity of the car B after the impact = ?
u₂ = velocity of the car B before the impact = 0 m/s (it was initially at rest)
u₁ = velocity of car A before the impact = ?
First of, we can solve for v₂, the velocity of car B after the impact, from some of the information given in the question.
- Skid marks indicate car B slid 10 m after the impact
- The coefficient of kinetic friction the tires and road is 0.8.
According to the work energy theorem, the work done by frictional force in stopping the car B is equal to the change in kinetic energy of the car B. (All after collision)
W = ΔK.E
ΔK.E = (1/2)(1200)(v₂²) - 0 (final kinetic energy is 0 since the car comes to stop eventually)
ΔK.E = (600v₂²) J
W = F × d
where F = frictional force = μmg = 0.8×1300×9.8 = 10,192 N
d = distance the car skids over before stopping = 10 m
W = 10,192 × 10 = 101,920 J
W = ΔK.E
101,920 = 600v₂²
v₂² = (101920/600) = 169.867
v₂ = 13.03 m/s
But recall,
Coefficient of restitution = (v₁ - v₂)/(u₂ - u₁)
For the sake of convention, we take the direction of car A's initial velocity to be the positive direction.
u₁ = ?
u₂ = 0 m/s
v₁ = ?
v₂ = +13.03 m/s
Coefficient of restitution = 0.4
0.4 = (v₁ - 13.03)/(0 - u₁)
-0.4u₁ = v₁ - 13.03
v₁ = 13.03 - 0.4u₁
But this is a collision. In a collision, the linear momentum is usually conserved.
Momentum before collision = Momentum after collision
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
1300u₁ + (1200×0) = 1300v₁ + (1200×13.03)
1300u₁ + 0 = 1300v₁ + 15639.95
1300u₁ = 1300v₁ + 15639.95
But recall, from the coefficient of restitution relation,
v₁ = 13.03 - 0.4u₁
Substituting this into the momentum balance equation.
1300u₁ = 1300v₁ + 15639.95
1300u₁ = 1300(13.03 - 0.4u₁) + 15639.95
1300u₁ = 16943.28 - 520u₁ + 15639.95
1820u₁ = 32,583.23
u₁ = (32,583.23/1820)
u₁ = 17.90 m/s
Therefore, the velocity of A before impact = 17.90 m/s
Hope this Helps!!!
