Suppose that a car can accelerate from 30 mph to 50 mph in 4 seconds. Assuming a constant acceleration, find the acceleration (in miles per second squared) of the car and find the distance traveled by the car during the 4 seconds.
30 mph = 30/3600 miles/sec = 1/120 mps
50 mph = 1/72 mps
v = at + c
when t = 0, v = 1/120, so c = 1/120
v = at + 1/120
when t = 4, v = 1/72
1/72 = 4a + 1/120
4a = 1/180
a = 1/720 miles/sec^2
v = (1/720)t + 1/120
if the distance is s
s = (1/1440)t^2 + (1/120)t + k
when t = 0, s = 0, ---> k=0
s = (1/1440)t^2 + (1/120)t
when t = 4
s = (1/1440)(16) + (1/120)(4) = 2/45 miles
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To find the acceleration of the car, we can use the formula:
acceleration = (final velocity - initial velocity) / time
In this case, the final velocity is 50 mph, the initial velocity is 30 mph, and the time is 4 seconds.
acceleration = (50 mph - 30 mph) / 4 s
Now, let's convert the units from mph to miles per second (mps).
1 mile = 1.60934 kilometers
1 hour = 3600 seconds
So, 1 mph = 1.60934 km / 3600 s
To convert mph to mps, we multiply by the conversion factor:
1 mph * 1.60934 km / 3600 s = 0.00044704 km/s
With this conversion, we can calculate the acceleration:
acceleration = (50 mph - 30 mph) * 0.00044704 km/s / 4 s
acceleration = 20 mph * 0.00044704 km/s / 4 s
acceleration ≈ 0.00022352 km/s²
Therefore, the acceleration of the car is approximately 0.00022352 km/s² (or you can convert km/s² to miles per second squared if required).
To find the distance traveled by the car during the 4 seconds, we can use the formula:
distance = initial velocity * time + (1/2) * acceleration * time²
In this case, the initial velocity is 30 mph, the time is 4 seconds, and the acceleration is 0.00022352 km/s².
Let's convert the initial velocity to mps using the conversion factor:
30 mph * 0.00044704 km/s ≈ 0.0134112 km/s
Now, let's substitute these values into the formula:
distance = 0.0134112 km/s * 4 s + (1/2) * 0.00022352 km/s² * (4 s)²
distance ≈ 0.0536448 km + 0.00178816 km
distance ≈ 0.055432 km
Therefore, the car traveled approximately 0.055432 km (or you can convert km to miles if required) during the 4 seconds.