As a traffic light turns green, a waiting car starts with a constant acceleration of 6.0 m/s^2.At the instant the car begins to accelerate, a truck with a constant velcoity of 21 m/s pass in the next lane.
(a) How far will the car travel before it overtakes the truck?
(b) How fast will the car be travelling when it overtakes the truck?
Calculations (a) :
Car:
Vi = 0 m/s
Vf = ?
Acceleration = 6.0 m/s^2
Truck:
Vi = 21 m/s
Vf = 21 m/s
Acceleration = 0 m/s^2
* in order to figure out (a), the deltaD's of both the car and truck must be equal
- i'm not to sure if the deltaT will be the same but it seems as though the intial time is the same
This is all I know so far; I've been trying to sub in equations but nothing seems to work. Could someone please guide me in the right direction. Thanks :)
Write equations for the distance travelled by each car since the instant that the truck passes the car. Set the two distances equal to one another and solve for t. Once you have that, use the equations for the position and velocity of the truck vs time.
Xt = 21 t
Xc = (1/2*6 t^2 = 3 t^2
Xc = Xt when 3t^2 = 21 t
t^2 - 7t = 0
t*(t-7) = 0
t = 0 or 7 s
t=0 is the start time when the truck passes the car; you want the other answer.
Vc = a t = 42 m/s when the car passes the truck
To solve this problem, you need to find the time it takes for the car to overtake the truck. Once you have the time, you can use that to find the distance traveled by the car and its velocity at that moment.
Let's start by calculating the time it takes for the car to catch up to the truck:
For the car:
Vi = 0 m/s (initial velocity)
Vf = ? (final velocity)
Acceleration = 6.0 m/s²
For the truck:
Vi = 21 m/s (constant velocity)
Vf = 21 m/s (constant velocity)
Acceleration = 0 m/s²
Since the car is accelerating and the truck is moving at a constant velocity, there will be a point where the car catches up to the truck. At this point, their displacements will be equal. We can use the following equation to find the time it takes for the car to catch up to the truck:
Car's displacement = Truck's displacement
Using the kinematic equation:
Car's displacement = Vi*t + 0.5*a*t²
Truck's displacement = Vi*t
where Vi is the initial velocity, a is the acceleration, t is the time, and the is the displacement.
Setting both displacements equal, we get:
0.5*6.0*t² = 21*t
Simplifying the equation, we have:
3.0*t² = 21*t
Dividing both sides by t, we get:
3.0*t = 21
Now, we can solve for t:
t = 21 / 3.0
t = 7 seconds
So, it will take the car 7 seconds to overtake the truck.
Now, let's calculate the distance the car travels and its velocity at that time:
Using the equation for displacement:
Car's displacement = Vi*t + 0.5*a*t²
Substituting the values:
Car's displacement = 0*t + 0.5*6.0*(7)²
Car's displacement = 0 + 0.5*6.0*49
Car's displacement = 0 + 147.0
Car's displacement = 147.0 meters
So, the car will travel a distance of 147.0 meters before overtaking the truck.
To find the car's velocity at that time, we can use the equation:
Vf = Vi + a*t
Substituting the values:
Vf = 0 + 6.0*7
Vf = 42.0 m/s
Therefore, the car will be traveling at a velocity of 42.0 m/s when it overtakes the truck.