A car sits in an entrance ramp to a freeway, waiting for a break in the traffic. The driver sees a small gap between a van and an 18-wheel truck and accelerates with constant acceleration along the ramp and onto the freeway. The car starts from rest, moves in a straight line, and has a speed of 15.0m/s when it reaches the end of the ramp, which has length 125m .
(a) What is the acceleration of the car?
(b) How much time does it take the car to travel the length of the ramp?
a. V^2 = Vo^2 + 2a*d
a=(V^2-Vo^2)/2d = (15^2-0)/250=0.9 m/s^2
b. V = Vo + a*t
t = (V-Vo)/a = (15-0)/0.9 = 16.67 s.
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(a) What is the acceleration of the car?
Well, I can see why the car wants to get on the freeway, I mean, who wouldn't? But let's get serious for a moment. We can solve this problem using the kinematic equation:
v² = u² + 2as
where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the displacement.
Since the car starts from rest (u = 0 m/s) and reaches a final velocity of 15.0 m/s, we can rearrange the equation to solve for the acceleration:
a = (v² - u²) / (2s)
Plugging in the values, we get:
a = (15.0² - 0) / (2 * 125)
a = 225.0 / 250
a = 0.9 m/s²
So, the acceleration of the car is 0.9 m/s².
(b) How much time does it take the car to travel the length of the ramp?
For this, we can use another kinematic equation:
v = u + at
where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time taken.
Again, since the car starts from rest (u = 0 m/s) and reaches a final velocity of 15.0 m/s, we can rearrange the equation to solve for time:
t = (v - u) / a
Plugging in the values:
t = (15.0 - 0) / 0.9
t = 16.7 seconds
So, it takes the car approximately 16.7 seconds to travel the length of the ramp. Just enough time to practice singing along to your favorite tunes!
To find the acceleration of the car, we can use the formula:
v² = u² + 2as
where:
v = final velocity of the car = 15.0 m/s
u = initial velocity of the car = 0 m/s (since it starts from rest)
a = acceleration of the car (unknown)
s = distance traveled by the car = 125 m (length of the ramp)
Simplifying the formula, we get:
a = (v² - u²) / (2s)
Substituting the values we know:
a = (15.0m/s)² - (0m/s)² / (2 * 125m)
Simplifying further, we get:
a = 225m²/s² / 250m
a = 0.9 m/s²
Therefore, the acceleration of the car is 0.9 m/s².
Now, to find the time it takes for the car to travel the length of the ramp, we can use the formula:
s = ut + 0.5at²
Since the car starts from rest, its initial velocity (u) is 0 m/s. Therefore, the formula simplifies to:
s = 0.5at²
Substituting the known values:
125m = 0.5 * (0.9m/s²) * t²
Simplifying further, we get:
125m = 0.45m/s² * t²
Dividing both sides of the equation by 0.45m/s²:
t² = 125m / 0.45m/s²
t² = 277.78s²
Taking the square root of both sides:
t = sqrt(277.78s²)
t ≈ 16.67s
Therefore, it takes approximately 16.67 seconds for the car to travel the length of the ramp.