A 0.80 10^3 kg Toyota collides into the rear end of a 2.6 1^03 kg Cadillac stopped at a red light. The bumpers lock, the brakes are locked, and the two cars skid forward 5.0 m before stopping. The police officer, knowing that the coefficient of kinetic friction between tires and road is 0.40, calculates the speed of the Toyota at impact. What is that speed?
m g down
mu m g friction force
F = m a
mu m g = m a
a = mu g = .4 (9.81) = 3.92 m/s^2
a = change in speed/time
change in speed = v
average speed during stop = v/2
distance = average speed * time
5 meters = v/2 t
so
t = 10/v
a = v /(10/v) = v^2/10
so
v^2/10 = 3.92
v^2 = 39.2
v = 6.26 m/s
Well, it seems like the Toyota and the Cadillac wanted to get up close and personal at the red light. Let's calculate the speed of the Toyota before this bumper-to-bumper encounter!
To start, we can use the conservation of momentum to solve this problem. The momentum before the collision is equal to the momentum after the collision (assuming there are no external forces acting on the system).
The momentum before the collision is given by:
p_before = m_toyota * v_toyota, where m_toyota is the mass of the Toyota and v_toyota is its velocity before the collision.
The momentum after the collision is zero, since both cars come to a stop.
The Toyota and the Cadillac skid together for 5.0 m, so the work done by the frictional force between the tires and the road is given by:
work = force_friction * distance
Since the cars come to a stop, the work done by the frictional force is equal to the change in kinetic energy, which is given by:
work = (1/2) * (m_toyota + m_cadillac) * v_after^2, where m_cadillac is the mass of the Cadillac and v_after is the final velocity of both cars.
Now, the frictional force can be calculated using the coefficient of kinetic friction, which is given by:
force_friction = coefficient_friction * normal_force.
The normal force is equal to the weight of the cars, which is given by:
normal_force = (m_toyota + m_cadillac) * g, where g is the acceleration due to gravity.
Combining all these equations and solving for v_toyota, we find:
m_toyota * v_toyota = sqrt(2 * coefficient_friction * (m_toyota + m_cadillac) * g * distance)
Now, let's plug in the values:
m_toyota = 0.80 10^3 kg
m_cadillac = 2.6 10^3 kg
coefficient_friction = 0.40
g ≈ 9.8 m/s^2
distance = 5.0 m
After doing all the necessary calculations, the speed of the Toyota at impact should be around...wait for it... voom voom...15.38 m/s!
To find the speed of the Toyota at impact, we can use the concept of conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision. The momentum is given by the formula:
Momentum (p) = mass (m) × velocity (v)
Let's assume the initial velocity of the Toyota is (v1) and the initial velocity of the Cadillac is 0.
The momentum before the collision is given by:
Initial momentum = (mass of Toyota × velocity of Toyota) + (mass of Cadillac × velocity of Cadillac)
Since the Cadillac is stopped, its initial velocity is 0:
Initial momentum = (mass of Toyota × velocity of Toyota) + (0 × 0)
The momentum after the collision is given by:
Final momentum = (mass of Toyota + mass of Cadillac) × final velocity
Since the cars skid forward and stop, the final velocity is 0:
Final momentum = (mass of Toyota + mass of Cadillac) × 0
According to the principle of conservation of momentum:
Initial momentum = Final momentum
Therefore, we can set up the equation:
(mass of Toyota × velocity of Toyota) = 0
Solving for the velocity of the Toyota (v1):
velocity of Toyota = 0 / mass of Toyota
Given that the mass of the Toyota is 0.80 × 10^3 kg:
velocity of Toyota = 0 / 0.80 × 10^3 kg
velocity of Toyota = 0 m/s
Therefore, the speed of the Toyota at impact is 0 m/s.
To calculate the speed of the Toyota at impact, we can use the principle of conservation of momentum. The total momentum before the collision will be equal to the total momentum after the collision.
The momentum of an object is defined as the product of its mass and velocity, given by the equation:
Momentum = mass * velocity
Before the collision:
The momentum of the Toyota before the collision is given by:
Momentum_Toyota_before = mass_Toyota * velocity_Toyota
The momentum of the Cadillac before the collision is given by:
Momentum_Cadillac_before = mass_Cadillac * velocity_Cadillac
Since the Cadillac is stopped, its initial velocity (velocity_Cadillac) is zero.
After the collision:
The momentum of the Toyota after the collision is still given by:
Momentum_Toyota_after = mass_Toyota * velocity_Toyota_at_impact
The momentum of the Cadillac after the collision is:
Momentum_Cadillac_after = mass_Cadillac * final_velocity_Cadillac
Since the bumpers lock and the two cars skid together, the final velocities of both cars will be the same.
According to the principle of conservation of momentum:
Total momentum before the collision = Total momentum after the collision
Momentum_Toyota_before + Momentum_Cadillac_before = Momentum_Toyota_after + Momentum_Cadillac_after
mass_Toyota * velocity_Toyota + mass_Cadillac * velocity_Cadillac = mass_Toyota * velocity_Toyota_at_impact + mass_Cadillac * final_velocity_Cadillac
Now, let's plug in the given values:
mass_Toyota = 0.80 * 10^3 kg
mass_Cadillac = 2.6 * 10^3 kg
velocity_Cadillac = 0 m/s (since it is stopped)
final_velocity_Cadillac = velocity_Toyota_at_impact (since the two cars skid together)
The equation is now:
(0.80 * 10^3 kg) * velocity_Toyota + (2.6 * 10^3 kg) * 0 m/s = (0.80 * 10^3 kg) * velocity_Toyota_at_impact + (2.6 * 10^3 kg) * velocity_Toyota_at_impact
Simplifying the equation, we get:
(0.80 * 10^3 kg) * velocity_Toyota = (0.80 * 10^3 kg + 2.6 * 10^3 kg) * velocity_Toyota_at_impact
Dividing both sides by (0.80 * 10^3 kg):
velocity_Toyota = (0.80 * 10^3 kg + 2.6 * 10^3 kg) * velocity_Toyota_at_impact / (0.80 * 10^3 kg)
Now, let's substitute the coefficient of kinetic friction and the distance traveled into the equation:
μ (coefficient of kinetic friction) = 0.40
distance_traveled = 5.0 m
The equation becomes:
velocity_Toyota = (0.80 * 10^3 kg + 2.6 * 10^3 kg) * velocity_Toyota_at_impact / (0.80 * 10^3 kg) - 2 * μ * g * distance_traveled
μ * g = 0.40 * 9.8 m/s^2 (where g is the acceleration due to gravity)
Now, we can calculate the velocity of the Toyota at impact using the given values.