a car and a truck start from rest at the same instant with the car initially 50m behind the truck. The truck has a constant acceleration of 2m/s^2 and the car has a constant acceleration of 3m/s^2.
The car overtakes the truck after how many seconds?
At what speed is the car moving when it overtakes the truck?
What is the speed of the truck when its overtaken by the car?
How far has the truck traveled in meters when the car overtakes it?
1)
Car -
S = 50 (50m to travel to reach truck since truck is moving away at a constant acceleration)
u = o
a = 3
t = ?
S = ut + 0.5*a*t^2
50 = 0 + 0.5(3)t^2
t = sqr(100/3) = 5.77s
2)
S = 50
u = o
v = ?
a = 3
t = 5.77
v = u + at
v = 0 + 3*5.77
v = 17.3 m/s
3)
S = -
u = o
v = ?
a = 2
t = 5.77
v = u + at
v = 0 + 2*5.77
v = 11.5 m/s
4)
S = ?
u = o
v = 11.5
a = 2
t = 5.77
S = ut + 0.5at^2
S = 0 + 0.5(2)(5.77^2)
S = 33.3 m
To find the time it takes for the car to overtake the truck, we can use the kinematic equation:
position = initial_position + (initial_velocity * time) + (0.5 * acceleration * time^2)
For the truck, its initial position is 0 (as it starts from rest) and its acceleration is 2 m/s^2. Therefore, the equation for the position of the truck is:
position_truck = (0.5 * 2 * time^2)
For the car, its initial position is -50 meters (as it starts 50 meters behind the truck) and its acceleration is 3 m/s^2. Therefore, the equation for the position of the car is:
position_car = -50 + (0.5 * 3 * time^2)
Since the car overtakes the truck, their positions will be equal when the car overtakes the truck. Solving for time:
(0.5 * 2 * time^2) = -50 + (0.5 * 3 * time^2)
0.5 * 2 * time^2 - 0.5 * 3 * time^2 = -50
time^2 = -50 / (0.5 * 2 - 0.5 * 3)
time^2 = -50 / (1 - 1.5)
time^2 = -50 / (-0.5)
time^2 = 100
time = √100
time = 10 seconds
So, the car overtakes the truck after 10 seconds.
To find the speed of the car when it overtakes the truck, we can use the formula:
speed = initial_velocity + (acceleration * time)
The car starts from rest, so its initial velocity is 0. Its acceleration is 3 m/s^2. Plugging in the values:
speed = 0 + (3 * 10)
speed = 30 m/s
So, the car is moving at a speed of 30 m/s when it overtakes the truck.
To find the speed of the truck when it is overtaken by the car, we can again use the formula:
speed = initial_velocity + (acceleration * time)
The truck starts from rest, so its initial velocity is 0. Its acceleration is 2 m/s^2. Plugging in the values:
speed = 0 + (2 * 10)
speed = 20 m/s
So, the speed of the truck when it is overtaken by the car is 20 m/s.
To find the distance traveled by the truck when the car overtakes it, we can use the equation:
distance = initial_velocity * time + (0.5 * acceleration * time^2)
The truck starts from rest, so its initial velocity is 0 and its acceleration is 2 m/s^2. Plugging in the values:
distance = 0 + (0.5 * 2 * 10^2)
distance = 0 + (0.5 * 2 * 100)
distance = 0 + 100
distance = 100 meters
So, the truck has traveled 100 meters when the car overtakes it.