A 1340-kg car traveling
east at 13.6 m/s (20 mi/h) has a head-on collision with a
1930-kg car traveling west at 20.5 m/s (30 mi/h). If the collision
time is 0.10 s, what is the force needed to restrain a 68-kg
person in the smaller car? In the larger car?
I understand how to find out what the magnitude and direction of collision will be, but I do not understand exactly what this question is asking- or how to find the answer. Thanks!
To solve this problem, we need to apply the principle of conservation of linear momentum. This principle states that the total linear momentum before a collision is equal to the total linear momentum after the collision, assuming no external forces are acting on the system.
The linear momentum of an object is given by the equation:
linear momentum (p) = mass (m) × velocity (v)
First, let's convert the speeds from mi/h to m/s:
Speed of the car traveling east: 13.6 m/s
Speed of the car traveling west: 20.5 m/s
Now, we can calculate the initial momentum of each car:
Initial momentum of the 1340-kg car (car A):
pA = m × v = 1340 kg × 13.6 m/s
Initial momentum of the 1930-kg car (car B):
pB = m × v = 1930 kg × (-20.5 m/s) (negative since the car is traveling west)
Next, let's calculate the final momentum of the system. Since the cars collide and come to a stop, their final velocities are both zero. Therefore, the final momentum of both cars is zero.
According to the principle of conservation of linear momentum:
Total initial momentum = Total final momentum
pA + pB = 0 + 0
Now, we can solve for the initial velocity of car B (before the collision):
pA = -pB
1340 kg × 13.6 m/s = 1930 kg × vb (vb is the velocity of car B before the collision)
Solving for vb:
vb = (1340 kg × 13.6 m/s) / 1930 kg
Now that we know the initial velocity of car B, we can find the change in velocity of the person in each car during the collision. This change in velocity will be the same for both cars, since the collision time is the same.
Change in velocity of the person in the smaller car:
Δv (person in smaller car) = vb - 0 (since the person starts at rest)
Finally, we can calculate the force needed to restrain the person in each car using the formula:
force (F) = mass (m) × acceleration (a)
Since the collision time (t) is given as 0.10 s, we can calculate acceleration using the equation:
acceleration (a) = Δv / t
Now, we can calculate the force needed to restrain the person in each car:
Force on the person in the smaller car:
F (person in smaller car) = 68 kg × (Δv / 0.10 s)
Force on the person in the larger car:
F (person in larger car) = mass of larger car × (Δv / t)
Now, you can substitute the calculated values into the equations to find the forces on the person in each car.