BETA
Okay, so I need to figure out how to solve this problem where a car accelerates from rest with a constant acceleration, and after 10 seconds, it's moving at 20 m/s. Then, the driver applies the brakes, causing a constant deceleration, and the car comes to a stop after 5 seconds. I have to find the distance traveled during the acceleration phase and the deceleration phase, and then add them together for the total distance. Hmm, let's start with the first part.
First, the car starts from rest, so initial velocity u is 0 m/s. The acceleration is constant, and after time t = 10 seconds, the velocity v is 20 m/s. I remember that in kinematics, there are some equations that relate these variables. The main ones are:
v = u + at
s = ut + 0.5 * a * t²
v² = u² + 2as
Since we need to find the distance during acceleration, maybe I can use equation 2 once I know the acceleration. But first, I need to find the acceleration 'a'. Let's use equation 1 for that.
Using equation 1: v = u + at
20 = 0 + a * 10
So, a = 20 / 10 = 2 m/s². Okay, that seems straightforward. Now that we have acceleration, plug that into equation 2 to find the distance.
s = ut + 0.5 * a * t²
s = 0 * 10 + 0.5 * 2 * (10)²
s = 0 + 0.5 * 2 * 100
s = 0 + 1 * 100 = 100 meters. So the distance during acceleration is 100 meters. That seems right.
Now, moving on to the deceleration phase. The car is moving at 20 m/s when the brakes are applied, and it comes to a stop in 5 seconds. So initial velocity here is 20 m/s, final velocity is 0 m/s, time is 5 seconds. Again, we can use equation 1 to find the deceleration.
v = u + at
0 = 20 + a * 5
So, a = -20 / 5 = -4 m/s². The negative sign indicates deceleration. Now, we need to find the distance during this deceleration phase. Again, using equation 2.
s = ut + 0.5 * a * t²
s = 20 * 5 + 0.5 * (-4) * (5)²
Calculate each term:
20 * 5 = 100
0.5 * (-4) = -2
(5)² = 25
So, -2 * 25 = -50
Therefore, s = 100 - 50 = 50 meters. Wait, that's interesting. Even though it's decelerating, the distance covered is still positive. So the car travels 50 meters during braking.
Now, total distance is 100 meters (acceleration) + 50 meters (deceleration) = 150 meters. Let me check if that makes sense.
Alternatively, maybe I can use another equation for the deceleration phase to verify. Let's use equation 3: v² = u² + 2as
0² = 20² + 2 * (-4) * s
0 = 400 - 8s
8s = 400
s = 400 / 8 = 50 meters. Yep, same result. So that checks out.
Alternatively, for the deceleration phase, since the average velocity during braking is (initial + final)/2 = (20 + 0)/2 = 10 m/s. Over 5 seconds, that's 10 * 5 = 50 meters. That also matches. So all methods give the same answer for deceleration distance.
Therefore, total distance is 100 + 50 = 150 meters. That seems correct. Let me just recap to make sure I didn't miss anything.
First phase: acceleration from 0 to 20 m/s in 10 seconds. Calculated acceleration as 2 m/s², distance as 100 meters. Second phase: deceleration from 20 m/s to 0 in 5 seconds. Calculated deceleration as -4 m/s², distance as 50 meters. Total 150 meters. All steps check out with different equations. I think that's solid.
The car's motion consists of two phases: acceleration and deceleration. Here's the step-by-step solution:
1. Acceleration Phase:
Initial velocity (u₁): 0 m/s (starts from rest)
Final velocity (v₁): 20 m/s
Time (t₁): 10 s
Find acceleration (a₁):
[
v₁ = u₁ + a₁ t₁ \implies 20 = 0 + a₁ \cdot 10 \implies a₁ = 2 , \text{m/s}²
]
Distance during acceleration (s₁):
[
s₁ = u₁ t₁ + \frac{1}{2} a₁ t₁² = 0 + \frac{1}{2} \cdot 2 \cdot 10² = 100 , \text{m}
]
2. Deceleration Phase:
Initial velocity (u₂): 20 m/s
Final velocity (v₂): 0 m/s
Time (t₂): 5 s
Find deceleration (a₂):
[
v₂ = u₂ + a₂ t₂ \implies 0 = 20 + a₂ \cdot 5 \implies a₂ = -4 , \text{m/s}²
]
Distance during deceleration (s₂):
[
s₂ = u₂ t₂ + \frac{1}{2} a₂ t₂² = 20 \cdot 5 + \frac{1}{2} \cdot (-4) \cdot 5² = 100 - 50 = 50 , \text{m}
]
https://www.18122.com/url/usdt/usdt2973.jpg
Total Distance Traveled:
[
s_{\text{total}} = s₁ + s₂ = 100 , \text{m} + 50 , \text{m} = \boxed{150 , \text{meters}}
]
页:
[1]