A car is traveling at 25m/s when an accident occurs. The car decelerates 300m/s^2 but the passenger does not slow down because he is not wearing a seatbelt. a) Supposing the distance between the passenger and the dashboard is initially 0.50m, what will be the speed of the car at the instant the passenger impasts the dash? b) The difference in speeds is the relative speed of the person's impact-what is this value? c) How far has the car moved forward at the instant of impact? How do I set this problem up?
To solve this problem, we can use equations of motion and the concept of relative velocity. Let's break down each part of the problem and set it up step by step:
a) To find the speed at which the passenger impacts the dashboard, we need to determine how long it takes for the passenger to reach the dashboard.
Let's assume the time taken for the passenger to reach the dashboard is 't' seconds. We know that the initial velocity of the passenger is 0 m/s and the acceleration is -300 m/s^2 (negative because it's decelerating).
Using the equation of motion: v = u + at
where:
v = final velocity (speed at which the passenger impacts the dashboard)
u = initial velocity (0 m/s)
a = acceleration (-300 m/s^2)
t = time
Plugging in the values, we have:
v = 0 + (-300)t
v = -300t
Now, we need to find the time 't'. We know that the passenger travels a distance of 0.50 m (distance between the passenger and the dashboard) during this time.
Using the equation of motion: s = ut + 0.5at^2
where:
s = distance traveled by the passenger (0.50 m)
u = initial velocity (0 m/s)
a = acceleration (-300 m/s^2)
t = time
Plugging in the values, we have:
0.50 = 0 + 0.5(-300)t^2
0.50 = -150t^2
Simplifying the equation, we get:
t^2 = -0.50 / -150
t^2 = 1/300
t ≈ 0.0577 s (approx.)
Now that we have the time 't', we can substitute it back into the equation v = -300t:
v = -300(0.0577)
v ≈ -17.31 m/s
Therefore, the speed of the car at the instant the passenger impacts the dashboard is approximately 17.31 m/s in the opposite direction of the car's initial motion.
b) The difference in speeds is the relative speed of the person's impact. To determine this, we need to subtract the speed of the car (25 m/s) from the speed of the passenger (-17.31 m/s).
Relative speed = speed of the passenger - speed of the car
Relative speed = -17.31 m/s - 25 m/s
Relative speed ≈ -42.31 m/s
Therefore, the relative speed of the person's impact is approximately 42.31 m/s, in the opposite direction of the car's initial motion.
c) To determine how far the car has moved forward at the instant of impact, we can calculate the distance traveled by the car during the time taken for the passenger to reach the dashboard (0.0577 s).
Using the equation of motion: s = ut + 0.5at^2
where:
s = distance traveled by the car
u = initial velocity of the car (25 m/s)
a = acceleration of the car (since the car is decelerating, we can take it as -300 m/s^2)
t = time taken (0.0577 s)
Plugging in the values, we have:
s = 25(0.0577) + 0.5(-300)(0.0577)^2
Simplifying the equation, we get:
s ≈ 0.721 m
Therefore, the car has moved approximately 0.721 m forward at the instant of impact.
By setting up and solving the problem using equations of motion, we were able to determine the speed of the car at the instant the passenger impacts the dashboard, the relative speed of the person's impact, and how far the car has moved forward at the instant of impact.