A car and a train move together along straight, parallel paths with the same constant cruising speed v(initial). At t=0 the car driver notices a red light ahead and slows down with constant acceleration -a(initial). Just as the car comes to a full stop, the light immediately turns green, and the car then accelerates back to its original speed v(initial) with constant acceleration a(initial). During the same time interval, the train continues to travel at the constant speed v(initial).
How much time does it take for the car to come to a full stop? t1 = ?
How much time does it take for the car to accelerate from the full stop to its original speed? t2 = ?
The train does not stop at the stoplight. How far behind the train is the car when the car reaches its original speed v(initial) again?
delta d = ?
A car and a train move together along straight, parallel paths with the same constant cruising speed v0. At t=0 the car driver notices a red light ahead and slows down with constant acceleration −a0. Just as the car comes to a full stop, the light immediately turns green, and the car then accelerates back to its original speed v0 with constant acceleration a0. During the same time interval, the train continues to travel at the constant speed v0.
To find the answers to these questions, we can use the equations of motion for uniformly accelerated motion. Let's go through each question one by one:
1. How much time does it take for the car to come to a full stop? t1 = ?
To find the time it takes for the car to come to a full stop, we can use the equation of motion:
v = u + at
Where:
- v is the final velocity (0 m/s since the car comes to a full stop)
- u is the initial velocity (v(initial))
- a is the acceleration (-a(initial))
- t is the time
Rearranging the equation, we have:
t = (v - u) / a
Substituting the known values, we have:
t1 = (0 - v(initial)) / (-a(initial))
2. How much time does it take for the car to accelerate from the full stop to its original speed? t2 = ?
To find the time it takes for the car to accelerate from the full stop to its original speed, we can again use the equation of motion:
v = u + at
Where:
- v is the final velocity (v(initial))
- u is the initial velocity (0 m/s since the car starts from a full stop)
- a is the acceleration (a(initial))
- t is the time
Rearranging the equation, we have:
t = (v - u) / a
Substituting the known values, we have:
t2 = (v(initial) - 0) / a(initial)
3. How far behind the train is the car when the car reaches its original speed (v(initial)) again? delta d = ?
To find the distance between the car and the train when the car reaches its original speed again, we need to determine the distance traveled during the deceleration period (when the car comes to a full stop) and the distance traveled during the acceleration period (when the car accelerates back to its original speed).
The distance traveled during uniform acceleration can be determined using the equation:
s = ut + (1/2)at^2
During the deceleration period, the initial velocity u is v(initial), the acceleration a is -a(initial), and the time t is t1 (the time it takes for the car to come to a full stop).
During the acceleration period, the initial velocity u is 0 m/s (since the car starts from a full stop), the acceleration a is a(initial), and the time t is t2 (the time it takes for the car to accelerate from the full stop to its original speed).
The total distance traveled by the car is the sum of the distances during the deceleration and acceleration periods.
delta d = s1 + s2 = (v(initial) * t1) + (0.5 * a(initial) * t2^2)
Substituting the previously calculated values of t1 and t2, we can find the distance delta d.
Please note that in all these equations, it is important to use consistent units for time, velocity, and acceleration.