One study suggests that a brain concussion can occur if a person's head is subjected to an acceleration as small as 60g, where g is the acceleration of gravity.
a. If the mass of the person's head is 4.5 kg, what is the magnitude of the force?
(Express your answer to two significant figures and include appropriate units.)
b. If the impulse delivered to the person's head is 19 kg⋅m/s, over what time interval is the force applied?
(Express your answer to two significant figures and include appropriate units.)
60 g is not a small force.
a. F=ma
Force=4.5*60*9.8 N
= 2646 N
b. Force*time=19
time= 19/2646
=0.00718 s
a. Well, well, well, looks like someone took a real "hit" with this question. Anyway, we'll have to do some calculations to figure this out. Now, we know that the acceleration is 60g, where g is the acceleration due to gravity.
So, we'll start by converting g into the actual acceleration value. We know g is approximately 9.8 m/s^2. Therefore, 60g would be 60 times that, which is 588 m/s^2.
Next, we need to find the force. We can use Newton's second law, which states that force (F) is equal to mass (m) times acceleration (a). The mass of the person's head is 4.5 kg, and the acceleration is 588 m/s^2.
Now it's just a matter of multiplying them together to find the force.
F = m * a
F = 4.5 kg * 588 m/s^2
F ≈ 2,646 N
So, the magnitude of the force is approximately 2,646 Newtons. Ouch!
b. Now, let's move on to part b. We have the impulse delivered to the person's head, which is given as 19 kg·m/s. Impulse is defined as the change in momentum of an object, which can be calculated using the equation:
Impulse = force * time
We can rearrange the equation to solve for time:
time = impulse / force
Plugging in the values we have:
time = 19 kg·m/s / 2,646 N
time ≈ 0.0072 s
So, the time interval over which the force is applied is approximately 0.0072 seconds. That's quite a "wham, bam, thank you, ma'am" situation, isn't it?
a. To find the magnitude of the force, we need to use Newton's second law of motion, which states that force (F) equals mass (m) multiplied by acceleration (a). In this case, the acceleration is given as 60g, where g is the acceleration due to gravity.
The acceleration due to gravity is approximately 9.8 m/s^2. Therefore, 60g is equal to 60 times 9.8 m/s^2.
Acceleration = 60g = 60 × 9.8 m/s^2 = 588 m/s^2 (rounded to two significant figures)
Now, we can calculate the magnitude of the force using F = m × a, where m is the mass of the person's head given as 4.5 kg.
Force (F) = mass (m) × acceleration (a)
F = 4.5 kg × 588 m/s^2
Calculating the force:
F = 2646 N (rounded to two significant figures)
Therefore, the magnitude of the force is 2646 N.
b. The impulse delivered to the person's head is given as 19 kg⋅m/s. Impulse is equal to the force applied multiplied by the time interval (t).
Impulse (I) = force (F) × time (t)
19 kg⋅m/s = 2646 N × t
We need to solve for time (t), so we rearrange the equation:
t = I / F
t = 19 kg⋅m/s ÷ 2646 N
Calculating the time interval:
t ≈ 0.0072 s (rounded to two significant figures)
Therefore, the force is applied over a time interval of approximately 0.0072 s.
a. To find the magnitude of the force, we can use Newton's second law of motion, which states that force (F) is equal to mass (m) multiplied by acceleration (a). In this case, the acceleration is given in terms of g.
The acceleration in terms of g, denoted as ag, can be calculated by multiplying the acceleration due to gravity (g ≈ 9.8 m/s^2) by the number of g's. So, ag = 60g.
Let's substitute the known values into the equation:
F = m * ag
F = 4.5 kg * 60g
Now, we can calculate the magnitude of the force:
F = 4.5 kg * 60 * 9.8 m/s^2
F ≈ 2,646 N
Therefore, the magnitude of the force is approximately 2,646 Newtons.
b. The impulse delivered to the person's head is given by the product of the force (F) and the change in time (Δt). Mathematically, impulse (J) is equal to force multiplied by time interval: J = F * Δt.
To find the time interval (Δt), we rearrange the equation:
Δt = J / F
Substituting the known values:
Δt = 19 kg⋅m/s / 2,646 N
Δt ≈ 0.0072 s
Therefore, the time interval over which the force is applied is approximately 0.0072 seconds.