The front 1.20 m of a 1,350-kg car is designed as a "crumple zone" that collapses to absorb the shock of a collision.
(a) If a car traveling 22.0 m/s stops uniformly in 1.20 m, how long does the collision last?
s
(b) What is the magnitude of the average force on the car?
N
(c) What is the magnitude of the acceleration of the car? Express the acceleration as a multiple of the acceleration of gravity.
g
.109
To find the answers to these questions, we can use the equations of motion and the principles of uniform acceleration. Let's break down each part:
(a) To determine the duration of the collision, we can use the equation:
v = u + at
where:
v = final velocity (0 m/s, since the car stops)
u = initial velocity (22.0 m/s)
a = acceleration
t = time
In this case, we need to solve for time (t) and we know that distance (s) is 1.20 m. Rearranging the equation, we have:
t = (v - u) / a
Since the car stops uniformly, the acceleration (a) will be negative (deceleration). Therefore:
t = (0 - 22.0) / a
To find the value of acceleration (a), we can use the equation:
v^2 = u^2 + 2as
where:
v = final velocity (0 m/s)
u = initial velocity (22.0 m/s)
a = acceleration
s = distance (1.20 m)
Substituting the known values, we have:
0^2 = 22.0^2 + 2a(1.20)
Simplifying:
0 = 484 + 2.4a
Rearranging the equation:
2.4a = -484
a = -484 / 2.4
Now we can substitute the value of acceleration (a) into the equation for time (t):
t = (0 - 22.0) / (-484 / 2.4)
t = 22.0 / (484 / 2.4)
Calculating the value:
t = 0.12 s
So, the collision lasts for 0.12 seconds.
(b) To find the magnitude of the average force exerted on the car, we can use Newton's second law of motion:
F = ma
where:
F = force
m = mass of the car (1,350 kg)
a = acceleration (-484 / 2.4 m/s^2)
Substituting the known values:
F = 1,350 kg * (-484 / 2.4 m/s^2)
Calculating the value:
F ≈ -273,800 N
Since we are looking for the magnitude (absolute value) of the force, the average force exerted on the car is approximately 273,800 N.
(c) Finally, to determine the magnitude of the acceleration of the car expressed as a multiple of the acceleration due to gravity (g), we can divide the acceleration by g.
g = 9.8 m/s^2
Acceleration = (-484 / 2.4) m/s^2
Magnitude of acceleration = |(-484 / 2.4) / 9.8|
Calculating the value:
Magnitude of acceleration ≈ 20 g
Therefore, the magnitude of the acceleration of the car is approximately 20 times the acceleration due to gravity.
To solve this problem, we will use the kinematic equations and the principles of linear motion. Let's begin with part (a).
(a) To find the duration of the collision, we can use the equation:
\[ v_f = v_i + a \cdot t \]
Where:
- \( v_f \) is the final velocity (0 m/s since the car stops)
- \( v_i \) is the initial velocity (22.0 m/s)
- \( a \) is the acceleration (which is constant during the collision)
- \( t \) is the time we want to find
Rearranging the equation, we get:
\[ t = \frac{v_f - v_i}{a} \]
Since \( v_f = 0 \) m/s and the car stops uniformly in 1.20 m, we can find the acceleration using the equation:
\[ v_f^2 = v_i^2 + 2 \cdot a \cdot d \]
Where:
- \( v_f \) is the final velocity (0 m/s)
- \( v_i \) is the initial velocity (22.0 m/s)
- \( a \) is the acceleration (which is constant during the collision)
- \( d \) is the distance traveled (1.20 m)
Rearranging the equation, we get:
\[ a = \frac{v_f^2 - v_i^2}{2 \cdot d} \]
Now let's substitute the given values into the equation and calculate \( a \):
\[ a = \frac{0^2 - (22.0 \, \text{m/s})^2}{2 \cdot 1.20 \, \text{m}} \]
(a) The collision duration, or the time it takes for the car to stop, is given by \( t = \frac{v_f - v_i}{a} \). Let's calculate \( t \):
\[ t = \frac{0 - 22.0 \, \text{m/s}}{\frac{0^2 - (22.0 \, \text{m/s})^2}{2 \cdot 1.20 \, \text{m}}} \]
(b) To find the average force on the car, we can use Newton's second law of motion:
\[ F = m \cdot a \]
Where:
- \( F \) is the force
- \( m \) is the mass of the car (1,350 kg)
- \( a \) is the acceleration during the collision
Let's substitute the given values into the equation and calculate the force:
\[ F = (1,350 \, \text{kg}) \cdot a \]
(c) Finally, to express the acceleration in terms of the acceleration due to gravity (g), we can divide the acceleration by the acceleration due to gravity:
\[ a_{g} = \frac{a}{g} \]
Now let's calculate the values step-by-step.