A man claims he can safely hold on to a 11.90 kg child in a head-on collision with a relative speed of 117. mi/h lasting for 0.140 s as long as he has his seat belt on.
(a) Find the magnitude of the average force needed to hold onto the child.
To find the magnitude of the average force needed to hold onto the child, we can use the principle of impulse.
Impulse is defined as the change in momentum of an object and can be calculated using the equation: Impulse = Force × Time.
The change in momentum of the child can be calculated using the equation: Change in momentum = Mass × Change in velocity.
First, we need to convert the relative speed from miles per hour to meters per second.
1 mile = 1609.34 meters, and 1 hour = 3600 seconds. So, 117 mi/h = (117 × 1609.34) / 3600 = 52.1 m/s (rounded to one decimal place).
The change in velocity is equal to the relative speed, so Δv = 52.1 m/s.
Now, we can calculate the change in momentum of the child:
Change in momentum = Mass × Change in velocity
= 11.90 kg × 52.1 m/s
= 620.99 kg·m/s (rounded to two decimal places).
Next, we need to find the impulse using the duration of the collision provided, which is 0.140 s.
Impulse = Force × Time
Rearranging the formula, we can find the force:
Force = Impulse / Time
= 620.99 kg·m/s / 0.140 s
= 4435.64 N (rounded to two decimal places).
Therefore, the magnitude of the average force needed to hold onto the child is approximately 4435.64 N.
To find the magnitude of the average force needed to hold onto the child, we can use Newton's second law of motion, which states that the force acting on an object is equal to the product of its mass and acceleration.
First, let's convert the mass of the child from kilograms to pounds, since the relative speed is given in miles per hour. We know that 1 kilogram is approximately equal to 2.205 pounds, so:
Mass of the child = 11.90 kg * 2.205 lb/kg
Mass of the child ≈ 26.24 lb
Next, let's convert the relative speed from miles per hour to meters per second, since the SI unit for acceleration is meters per second per second. We know that 1 mile is equal to 1609.34 meters, and 1 hour is equal to 3600 seconds, so:
Relative speed = 117 mi/h * 1609.34 m/mi / 3600 s/h
Relative speed ≈ 52.36 m/s
Now, we can calculate the acceleration using the formula:
Acceleration = Change in velocity / Time
The change in velocity is the relative speed because the child is initially at rest. The time is given as 0.140 seconds, so:
Acceleration = 52.36 m/s / 0.140 s
Acceleration ≈ 374.00 m/s^2
Now that we have the mass and acceleration, we can use Newton's second law to calculate the force:
Force = Mass * Acceleration
Force = 26.24 lb * 374.00 m/s^2
Force ≈ 9,800 lb·ft/s^2
However, the unit of force in the SI system is Newtons (N), so let's convert the force back to Newtons:
1 lb·ft/s^2 ≈ 4.448 N
Force ≈ 9,800 lb·ft/s^2 * 4.448 N / 1 lb·ft/s^2
Force ≈ 43,470 N
Therefore, the magnitude of the average force needed to hold onto the child is approximately 43,470 Newtons.