A man claims he can safely hold on to a 12.80 kg child in a head-on collision with a relative speed of 133. mi/h lasting for 0.100 s as long as he has his seat belt on.
(a) Find the magnitude of the average force needed to hold onto the child.
133 mi/h=59.5 m/s
F=m•Δv/Δt=12.8•59.5/0.1=7616 N
To find the magnitude of the average force needed to hold onto the child, we can use the impulse-momentum principle.
The impulse-momentum principle states that the change in momentum of an object equals the impulse applied to it. Mathematically, this can be represented as:
Impulse = Change in Momentum
Impulse (J) is defined as the force applied to an object multiplied by the time interval over which the force is applied. It can be calculated using the formula:
J = F * Δt
Where,
F is the force applied to the object
Δt is the time interval over which the force is applied
The change in momentum (Δp) can be calculated using the formula:
Δp = m * Δv
Where,
m is the mass of the object
Δv is the change in velocity of the object
In this case, the mass of the child (m) is 12.80 kg, and the relative speed (Δv) is given as 133 mi/h. However, we need to convert the relative speed from miles per hour to meters per second to be consistent with the SI unit system.
1 mile is approximately equal to 1.60934 kilometers, and 1 kilometer is equal to 1000 meters. Therefore:
133 mi/h * 1.60934 km/mi * 1000 m/km * (1/3600) h/s ≈ 59.4 m/s
Now, we can calculate the change in momentum (Δp):
Δp = m * Δv
Δp = 12.80 kg * 59.4 m/s ≈ 762.72 kg·m/s
Since we are given the time interval (Δt) as 0.100 s, we can substitute the values into the impulse formula:
J = F * Δt
762.72 kg·m/s = F * 0.100 s
Now, solving for the force (F):
F = 762.72 kg·m/s / 0.100 s
F ≈ 7627.2 N
Therefore, the magnitude of the average force needed to hold onto the child is approximately 7627.2 Newtons.
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 mass of the object multiplied by its acceleration.
The acceleration can be found using the formula:
acceleration = change in velocity / time taken
The change in velocity can be calculated by converting the relative speed from miles per hour (mph) to meters per second (m/s). We know that 1 mile is equal to 1609.34 meters, and 1 hour is equal to 3600 seconds. Therefore, the relative speed can be converted as follows:
relative speed = 133 mi/h * 1609.34 m/1 mi * 1 h/3600 s
Next, we calculate the acceleration:
acceleration = (relative speed) / (time taken)
For time, we are given that it lasts for 0.100 seconds. Plugging in the values:
acceleration = (133 mi/h * 1609.34 m/1 mi * 1 h/3600 s) / 0.100 s
Now, let's calculate the acceleration.
acceleration = (133 * 1609.34) / (3600 * 0.100) m/s^2
Finally, we can calculate the force needed to hold onto the child using Newton's second law:
force = mass * acceleration
Given that the mass of the child is 12.80 kg, we can calculate the force:
force = 12.80 kg * acceleration
Now, compute the answer.