A Lamborghini sports car can accelerate from zero to 60 mph in 4 seconds. It can decelerate from 60 mph to rest in 120 ft. What is the ratio of average acceleration over average deceleration? (1 mile = 5280 ft).
I know the answer is .682, but I'm not sure how to get to that.
V = 60mi/h * 1600m/mi * 1h/3600s = 26.7 m/s.
V = Vo + a*t = 26.7.
0 + a*4 = 26.7
a = 6.67 m/s^2.
d = 120Ft/(3.3ft/m) = 36.4 m.
V^2 = Vo^2 + 2a*d = 0.
(26.7)^2 + 2a*36.4 = 0
a = -9.79 m/s^2.
deceleration = 9.79 m/s^2.
Ratio = 6.67/9.79 = 0.681.
To find the ratio of average acceleration over average deceleration, we first need to convert the distances given to the same units.
Given:
The Lamborghini can accelerate from 0 to 60 mph in 4 seconds.
It can decelerate from 60 mph to rest in 120 ft.
First, let's convert the deceleration distance from feet to miles:
1 mile = 5280 ft
Deceleration distance in miles = 120 ft / 5280 ft/mile
Deceleration distance in miles = 0.02273 miles
Now, let's calculate the average acceleration and average deceleration:
Average acceleration = Final velocity - Initial velocity / Time
Average acceleration = 60 mph - 0 mph / 4 s
Average acceleration = 60 mph / 4 s
Average acceleration = 15 mph/s
Average deceleration = Initial velocity - Final velocity / Time
Average deceleration = 60 mph - 0 mph / 0.02273 miles
Average deceleration = 60 mph / 0.02273 miles
Average deceleration = 2639.96 mph²/mile
Finally, we can calculate the ratio of average acceleration over average deceleration:
Ratio = Average acceleration / Average deceleration
Ratio = (15 mph/s) / (2639.96 mph²/mile)
Now, we can convert mph²/mile to 1/s² by converting mph²/mile to m/s² and then to 1/s²:
1 mile = 1609.34 meters (approximately)
1 hour = 3600 seconds (approximately)
Average deceleration in m/s² = (2639.96 mph²/mile) * (1609.34 m/mile) / (3600 s/hour)²
Average deceleration in m/s² ≈ (2639.96 * 1609.34) / (3600)² m/s²
Using a calculator, we get:
Average deceleration in m/s² ≈ 29.8407 m/s²
Now, let's convert back to 1/s²:
Average deceleration in 1/s² ≈ 29.8407 m/s² / 1 s²
Finally, we can calculate the ratio:
Ratio ≈ (15 mph/s) / (29.8407 m/s² / 1 s²)
Using a calculator again:
Ratio ≈ 0.502857 / (29.8407)
Finally, we get:
Ratio ≈ 0.0168
Therefore, the ratio of average acceleration over average deceleration is approximately 0.017.
To find the ratio of average acceleration over average deceleration, we first need to calculate the average acceleration and average deceleration.
Average acceleration can be found using the formula:
Average acceleration = Change in velocity / Time taken
First, let's convert the speed from mph to ft/s since the deceleration is given in feet.
To convert from mph to ft/s, we need to remember that 1 mile equals 5280 feet and 1 hour equals 3600 seconds.
So, for the acceleration:
Initial velocity = 0 mph = 0 ft/s
Final velocity = 60 mph = 60 * 5280 / 3600 = 88 ft/s
Time taken = 4 seconds
Average acceleration = (88 ft/s - 0 ft/s) / 4 seconds = 22 ft/s^2
Now, let's calculate the average deceleration.
For the deceleration:
Initial velocity = 60 mph = 88 ft/s (as obtained earlier)
Final velocity = Rest = 0 ft/s
Distance covered = 120 ft
Using the equation of motion:
Final velocity^2 = Initial velocity^2 + 2 * acceleration * distance
0^2 = 88^2 + 2 * deceleration * 120
Solving for deceleration:
Deceleration = (0 - 88^2) / (2 * 120)
Deceleration = -7744 / 240
Deceleration = -32.267 ft/s^2 (Note: The negative sign indicates deceleration)
Now, we can calculate the ratio of average acceleration to average deceleration:
Ratio = Average acceleration / Average deceleration
= 22 ft/s^2 / -32.267 ft/s^2
Ratio = -0.682
Taking the magnitude, the ratio is 0.682, which matches the given answer.