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 velocity of 21 m/s passes 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?
Write equations for the distance from the traffic light vs time for each car. Measure time from when they leave the light.
X1 = 3 t^2
X2 = 21 t
Set X1 = X2 and solve for t. Take the nonzero solution. Use that t and either equation to get the location. The velocity of the car at that time will be 6.0 * t. I expect it will be 42 m/s, since it will have to have the same average velocity as the truck at that time.
36.75
To solve this problem, we can use the equations of motion. Let's determine the time it takes for the car to catch up with the truck first.
A) Calculate the time it takes for the car to catch up with the truck:
1. Find the acceleration of the car: a = 6.0 m/s^2
2. Find the velocity of the truck: v_truck = 21 m/s
Using the formula: v = u + at, where v is the final velocity, u is the initial velocity, a is acceleration, and t is time:
u_car = 0 m/s (as the car starts from rest)
v_truck = 21 m/s
a_car = 6.0 m/s^2
v_car = u_car + a_car * t
21 m/s = 0 m/s + 6.0 m/s^2 * t
Now solve for t:
21 m/s = 6.0 m/s^2 * t
t = 21 m/s / 6.0 m/s^2
= 3.5 s
The car will take 3.5 seconds to catch up with the truck.
B) Calculate the speed of the car when it overtakes the truck:
Using the formula: v = u + at:
v_car = u_car + a_car * t
v_car = 0 m/s + 6.0 m/s^2 * 3.5 s
v_car = 21 m/s
The car will be traveling at a speed of 21 m/s when it overtakes the truck.
To find the answers to these questions, we need to apply the equations of motion.
Let's start with Question A: How far will the car travel before it overtakes the truck?
Step 1: Find the time it takes for the car to overtake the truck.
We can use the equation of motion: v = u + at, where
v is the final velocity of the car (when it overtakes the truck),
u is the initial velocity of the car,
a is the acceleration of the car, and
t is the time taken.
Given:
- Initial velocity of the car, u = 0 m/s (as it starts from rest)
- Acceleration of the car, a = 6.0 m/s^2
- Velocity of the truck, which is also the final velocity of the car when it overtakes the truck, v = 21 m/s
Plugging these values into the equation, we get:
21 m/s = 0 m/s + 6.0 m/s^2 * t
Simplifying, we have: 21 m/s = 6.0 m/s^2 * t
Step 2: Solve for time (t).
Rearranging the equation, we have:
t = 21 m/s / 6.0 m/s^2 ≈ 3.5 seconds
So it will take approximately 3.5 seconds for the car to overtake the truck.
Step 3: Find the distance traveled by the car during this time.
We can use the equation of motion: s = ut + 0.5 at^2, where
s is the distance traveled by the car,
u is the initial velocity of the car,
a is the acceleration of the car, and
t is the time taken.
Plugging in the values, we have:
s = 0 m/s * 3.5 s + 0.5 * 6.0 m/s^2 * (3.5 s)^2
Simplifying, we get:
s = 0 + 0.5 * 6.0 m/s^2 * 12.25 s^2
s ≈ 36.75 meters
Therefore, the car will travel approximately 36.75 meters before overtaking the truck.
Moving on to Question B: How fast will the car be traveling when it overtakes the truck?
We already found the time it takes for the car to overtake the truck, which is 3.5 seconds.
Using the equation of motion: v = u + at, where
v is the final velocity of the car (when it overtakes the truck),
u is the initial velocity of the car,
a is the acceleration of the car, and
t is the time taken.
Plugging in the values, we have:
v = 0 m/s + 6.0 m/s^2 * 3.5 s
Simplifying, we get:
v = 21 m/s
Therefore, the car will be traveling at a speed of 21 m/s when it overtakes the truck.