A car is stationary at a stop sign when it is hit directly from behind by a truck of mass 3000kg which was travelling at a speed of 9 m/s immediately before the collision. The two vehicles lock together and move forward with an initial speed of 6 m/s.
a) what is the mass of the car?
b) If the car had been designed with a crumple zone, explain in correct physics term, why this might reduce the risk of injury to the driver.
a) To find the mass of the car, we can use the principle of conservation of momentum.
The initial momentum of the truck before the collision can be calculated as the product of its mass (3000 kg) and velocity (9 m/s), which is:
Initial momentum (truck) = 3000 kg * 9 m/s = 27000 kg*m/s
After the collision, the two vehicles lock together and move forward with an initial velocity of 6 m/s. The total momentum of the system after the collision is the product of the combined mass of the two vehicles (m_car + 3000 kg) and the final velocity (6 m/s):
Final momentum (system) = (m_car + 3000 kg) * 6 m/s
According to the conservation of momentum, the initial and final momentum should be equal. So we can set up the equation:
Initial momentum (truck) = Final momentum (system)
27000 kg*m/s = (m_car + 3000 kg) * 6 m/s
Simplifying the equation:
27000 kg*m/s = 6 m/s * m_car + 6 m/s * 3000 kg
27000 kg*m/s = 6 m_car + 18000 kg*m/s
Subtracting 18000 kg*m/s from both sides:
9000 kg*m/s = 6 m_car
Dividing both sides by 6 m/s:
1500 kg = m_car
Therefore, the mass of the car is 1500 kg.
b) If the car had been designed with a crumple zone, it could reduce the risk of injury to the driver by absorbing some of the energy from the collision.
During a collision, the crumple zone, which is typically located at the front of the car, is designed to deform and collapse in a controlled manner. This deformation absorbs some of the kinetic energy of the impact, thereby reducing the force experienced by the driver and passengers.
By extending the duration of the collision, the crumple zone increases the time over which the change in momentum occurs. This reduces the acceleration of the occupants and decreases the likelihood of severe injuries, particularly to crucial body parts like the head and chest.
In other words, the crumple zone helps to slow down the rate at which the occupants experience a change in momentum, which reduces the likelihood of serious injury by reducing the force exerted on their bodies.
To find the mass of the car, we can use the principle of conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision.
Let's assume the mass of the car is "m".
a) Before the collision, the truck has a mass of 3000 kg and a velocity of 9 m/s. The car is stationary, so its velocity is 0 m/s.
Momentum before collision = (Mass of truck) * (Velocity of truck) + (Mass of car) * (Velocity of car)
3000 kg * 9 m/s + m * 0 m/s
After the collision, the truck and car lock together. They move forward with an initial speed of 6 m/s.
Momentum after collision = (Mass of truck + Mass of car) * (Velocity after collision)
(Mass of truck + m) * 6 m/s
According to the conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.
3000 kg * 9 m/s = (Mass of truck + m) * 6 m/s
27000 kg m/s = (3000 kg + m) * 6 m/s
Divide both sides of the equation by 6 m/s:
4500 kg = 3000 kg + m
Subtract 3000 kg from both sides of the equation:
m = 4500 kg - 3000 kg
m = 1500 kg
Therefore, the mass of the car is 1500 kg.
b) The purpose of a crumple zone in a car is to absorb and distribute the force of impact during a collision. When a collision occurs, the crumple zone, which is typically located in the front of the car, deforms and crushes. This deformation allows the crumple zone to absorb a significant amount of the kinetic energy generated during the impact.
By absorbing the energy and deforming, the crumple zone increases the time over which the collision takes place. This extends the duration of the impact and reduces the peak force acting on the occupants of the car. According to Newton's second law, force is equal to the change in momentum divided by the time taken for the change.
By increasing the time over which the collision occurs, the force acting on the car occupants is reduced. This decreases the risk of injury to the driver and passengers as the lower force reduces the deceleration experienced by their bodies. In other words, the crumple zone helps to slow down the car more gradually, spreading out the force and reducing the chances of severe impact-related injuries.