a) A set of crash tests consists of running a test car moving at a speed of 12.6 m/s (27.7 m/h) into a solid wall. Strapped securely in an advanced seat belt system, a 69.0 kg (151.8 lbs) dummy is found to move a distance of 0.600 m from the moment the car touches the wall to the time the car is stopped. Calculate the size of the average force which acts on the dummy during that time.
b) Using the direction of motion as positive direction, calculate the average acceleration of the dummy during that time (in g's) (use 1g=9.8 m/s2).
The answer to a is 9129 J but I am not sure of b
a) To calculate the average force acting on the dummy, we can use the work-energy principle. The work done on an object is equal to the force applied multiplied by the distance traveled.
The work done on the dummy can be calculated using the formula:
Work = Force * Distance
Given:
Speed of the car (v) = 12.6 m/s
Distance traveled by the dummy (d) = 0.600 m
Mass of the dummy (m) = 69.0 kg
First, we need to calculate the initial kinetic energy of the car:
Initial Kinetic Energy = 0.5 * mass * velocity^2
= 0.5 * 69.0 kg * (12.6 m/s)^2
Next, we calculate the work done on the dummy:
Work = Final Kinetic Energy - Initial Kinetic Energy
Since the car is brought to a stop, the final kinetic energy is zero.
Work = - Initial Kinetic Energy
Finally, we divide the work done by the distance traveled by the dummy to calculate the average force:
Average Force = Work / Distance
Let's calculate all the values and obtain the average force:
Initial Kinetic Energy = 0.5 * 69.0 kg * (12.6 m/s)^2
= 0.5 * 69.0 kg * 158.76 m^2/s^2
= 5444.38 J
Work = -5444.38 J
Average Force = Work / Distance
= -5444.38 J / 0.600 m
≈ -9064.03 N
The size of the average force acting on the dummy is approximately 9064.03 Newtons in the opposite direction to the motion.
b) Now let's calculate the average acceleration of the dummy in terms of g's.
Acceleration is defined as the change in velocity divided by the time taken:
Acceleration = Δv / Δt
Given the initial velocity (v) = 12.6 m/s and the final velocity (0 m/s), the change in velocity (Δv) is:
Δv = Final Velocity - Initial Velocity
= 0 m/s - 12.6 m/s
= -12.6 m/s (negative sign indicates deceleration)
Also, the time (Δt) is not provided directly. However, since we have already calculated the average force in the previous step, we can use Newton's second law of motion to find the time:
Average Force = mass * acceleration
Rearranging the equation, we have:
Acceleration = Average Force / mass
Acceleration = (-9064.03 N) / (69.0 kg)
≈ -131.3 m/s^2
Now, to convert this value into g's, we divide the acceleration by the acceleration due to gravity:
Average Acceleration = (-131.3 m/s^2) / (9.8 m/s^2)
≈ -13.4 g's
The average acceleration of the dummy during that time is approximately -13.4 g's, indicating deceleration. The negative sign indicates that the acceleration is in the opposite direction to the motion.
To calculate the size of the average force acting on the dummy during the crash test, we can make use of the work-energy principle.
a) The work done on an object can be calculated using the formula:
Work = Force × Distance × cosθ,
where θ is the angle between the direction of the force and the direction of motion. In this case, since the dummy is moving directly towards the wall, the angle between the force and the direction of motion is 0 degrees. Therefore, cosθ = 1.
The work done on the dummy is equal to its change in kinetic energy. The initial kinetic energy of the dummy can be calculated using the formula:
Initial Kinetic Energy = 0.5 × mass × initial velocity^2.
Plugging in the values:
Initial Kinetic Energy = 0.5 × 69.0 kg × (12.6 m/s)^2.
The final kinetic energy of the dummy is zero since it comes to rest. Therefore, the work done on the dummy is equal to its initial kinetic energy.
Work = Final Kinetic Energy - Initial Kinetic Energy.
Solving for the work done:
Work = 0 - Initial Kinetic Energy.
Now, we can calculate the average force using the equation:
Work = Force × Distance.
Rearranging the equation to solve for force:
Force = Work / Distance.
Plugging in the values:
Force = (0.5 × 69.0 kg × (12.6 m/s)^2) / 0.600 m.
Evaluating the expression:
Force = 9129 N.
Therefore, the size of the average force acting on the dummy during the crash test is 9129 N.
b) To calculate the average acceleration of the dummy during the crash test, we can use the equation:
Acceleration = (Final Velocity - Initial Velocity) / Time.
Since the final velocity is zero (the dummy comes to rest), the equation simplifies to:
Acceleration = - Initial Velocity / Time.
The initial velocity of the dummy can be calculated by converting the speed given in m/s to m/s by multiplying it by (1 hour / 60 minutes) and then (1 minute / 60 seconds):
Initial Velocity = 12.6 m/s × (1 hour / 60 minutes) × (1 minute / 60 seconds).
Calculating the initial velocity:
Initial Velocity = 12.6 m/s × (1/3600).
The time it takes for the dummy to come to rest is unknown. However, we know the distance the dummy moves during that time, which is 0.600 m.
To find time, we can use the equation of motion:
Distance = Initial Velocity × Time + 0.5 × Acceleration × Time^2.
Since the final velocity is zero, the equation simplifies to:
Distance = Initial Velocity × Time.
Rearranging the equation to solve for time:
Time = Distance / Initial Velocity.
Plugging in the values:
Time = 0.600 m / Initial Velocity.
Substituting the previously calculated value of initial velocity into the equation:
Time = 0.600 m / (12.6 m/s × (1/3600)).
Evaluating the expression:
Time = 17.8571 seconds.
Now, we can calculate the average acceleration:
Acceleration = - Initial Velocity / Time.
Plugging in the values:
Acceleration = (-12.6 m/s × (1/3600)) / 17.8571 seconds.
Evaluating the expression and converting to g's:
Acceleration = - (0.0035 m/s^2) / 17.8571 seconds × (1 g / 9.8 m/s^2).
Acceleration = - 0.0001993 g.
Therefore, the average acceleration of the dummy during the crash test is approximately -0.0002 g.