A truck is traveling with a uniform velocity 24 m/s, when it passes a car at rest, at this instant the car starts to move and accelerate at the rate of 6m/s^2 in the same direction as the truck.
a.) How long will it take the car to overtake the truck
b.) How fast will the car travel before it overtakes the truck.
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To find the answers to the questions, we need to determine the point at which the car catches up with the truck. Let's break it down step by step.
a.) How long will it take the car to overtake the truck?
To solve for the time it takes for the car to overtake the truck, we can set up an equation using the given information.
Let's first define the variables:
Vt = velocity of the truck = 24 m/s
Vc = velocity of the car
a = acceleration of the car = 6 m/s^2
t = time taken
We can determine the relationship between the distances traveled by the car and the truck using the formula:
Distance = Velocity × Time
For the truck:
Distance1 = Vt × t
For the car:
Distance2 = Vc × t + (1/2) × a × t^2
Now, since the car starts from rest, we know that initially, its velocity is 0, so we can modify the equation:
Distance2 = (1/2) × a × t^2
When the car overtakes the truck, the distances are equal:
Distance1 = Distance2
Vt × t = (1/2) × a × t^2
Simplifying the equation:
24t = (1/2) × 6 × t^2
24t = 3t^2
8t = t^2
t^2 - 8t = 0
t(t - 8) = 0
This quadratic equation has two solutions: t = 0 and t = 8.
Since time cannot be negative, we discard t = 0. Therefore, the car will take 8 seconds to overtake the truck.
b.) How fast will the car travel before it overtakes the truck?
To find the speed of the car when it overtakes the truck, we can substitute the value of t in either the equation Distance1 = Vt × t or Distance2 = (1/2) × a × t^2. Let's use the second equation:
Distance2 = (1/2) × a × t^2
Distance2 = (1/2) × 6 × (8)^2
Distance2 = 96 m
Now, we can use the equation for the car's velocity to find the speed:
Distance2 = Vc × t + (1/2) × a × t^2
96 = Vc × 8 + (1/2) × 6 × (8)^2
96 = 8Vc + 24(8)
96 = 8Vc + 192
8Vc = 96 - 192
8Vc = -96
Vc = -12 m/s
Since the velocity of a car cannot be negative in this case, it means that it has not yet overtaken the truck. Therefore, the speed of the car before it overtakes the truck is -12 m/s.