Question

The bumper of a new car is being tested. The \(2300-\mathrm{kg}\) vehicle, moving at \(15 \mathrm{~m} / \mathrm{s}\), is allowed to collide with a bridge abutment, being brought to rest in a time of \(0.54 \mathrm{~s}\). Find the average force that acted on the car during impact.

Short answer

The average force that acted on the car during the impact is -63894 N.

Step-by-step solution

Identify given values

The mass of the vehicle (\(m\)) is given as 2300 kg. The initial speed of the vehicle (\(v_i\)) is given as 15 m/s. The final speed of the vehicle (\(v_f\)) is 0 m/s, since it is brought to rest. The time (\(t\)) during which the car is brought to rest is given as 0.54 seconds.

Calculate the acceleration

The acceleration (\(a\)) can be calculated using the formula for acceleration, which is: \(a = \frac{{v_f - v_i}}{t}\). By substituting the given values into this equation, the acceleration can be calculated as \(a = \frac{{0 - 15}}{0.54} = -27.78 \, \mathrm{m/s^2}\). Negative sign indicates deceleration.

Calculate the force

The average force (\(F\)) that acted on the car during impact can be calculated using Newton's second law: \(F = m \cdot a\). Substituting the values of mass and acceleration into the equation, we get the force as \(F = 2300 \cdot -27.78 = -63894 \, \mathrm{N}\). The negative sign indicates that the force acted in the opposite direction to the car's initial motion.

Key concepts

average force

When analyzing the motion of objects in physics, particularly during collisions, the concept of average force is crucial.
In essence, average force provides a way to quantify the effect of forces over the duration of an event rather than at a single moment.

The average force can be understood as:

• Acting over a period when an object is in motion.
• Causing a change in velocity, like bringing a car to rest after a crash.
• Represented as a constant force that has the same effect on the motion as the real variable force.

Newton's Second Law is used to calculate average force, represented by the formula:\( F = m \cdot a \)Where \( F \) is the force, \( m \) is the mass, and \( a \) is the acceleration.
In a collision scenario where an abrupt stop occurs, the average force becomes negative, indicating the force opposes the motion.
This is crucial in automotive safety, where knowing the impact forces helps design better safety mechanisms.

deceleration

Deceleration refers to the reduction in speed or velocity of an object, equivalent to negative acceleration.
When an object slows down, it experiences deceleration, which is a reduction in kinetic energy.

In the context of collision:

• The initial speed of an object decreases rapidly due to an external force.
• This rapid decrease is what we calculate as deceleration.
• Unlike acceleration which is positive, deceleration is portrayed by a negative value.

The formula for calculating deceleration when given initial and final velocities is:\[a = \frac{v_f - v_i}{t}\]
Where \(a\) is the deceleration, \(v_f\) is the final velocity (0 if it comes to rest), \(v_i\) is the initial velocity, and \(t\) is the time duration.Deceleration is widely applicable to understand everyday phenomena, such as when a car stops at a red light or ceases movement due to a crash.

collision physics

Understanding collision physics is fundamental in analyzing impacts and their effects on objects.
Collisions are events where two or more bodies exert forces upon each other for a relatively short time.

Key points about collision physics include:

• Momentum change: The primary outcome during a collision.
• Force impact: During collision, forces acting are usually vast and act for a brief period.
• Energy conversion: Collisions often convert kinetic energy into other forms — heat, sound, and sometimes deformation.

In the scenario with the car hitting a bridge abutment: - The initial momentum of the car is significantly altered as it's brought to rest. - The deceleration induced by the collision reduces the car's speed to zero over a short duration. - By analyzing these collisions, engineers and physicists can design safer vehicles that better absorb impact forces, thus improving safety features like airbags and crumple zones.