A motorcycle is following a car that is traveling at constant speed on a straight highway. Initially, the car and the motorcycle are both traveling at the same speed of 21.5 m/s , and the distance between them is 57.0 m . After t1 = 3.00 s , the motorcycle starts to accelerate at a rate of 8.00 m/s2 . The motorcycle catches up with the car at some time t2.
They start out at the same speed so we are just on a surface moving at 21.5 m/s
All the motorcycle has to do is cover the difference, 57 meters
d = (1/2) a t^2
57 = (1/2) 8 * t^2
t^2 = 57/4 seconds to catch up (about 14 seconds). t2= t1 + t
the car travels 21.5 ( 3 + t) meters
and the motorcycle covers 57 meters more.
To find the time t2 when the motorcycle catches up with the car, we need to determine the distance traveled by both the car and the motorcycle after time t1 and set them equal to each other.
First, we can find the distance traveled by the car after t1 using the formula:
Distance = Speed x Time
Distance_car = Speed_car x Time
Distance_car = 21.5 m/s x 3.00 s
Distance_car = 64.5 m
Next, we need to find the distance traveled by the motorcycle after t1. Since the motorcycle starts accelerating after t1, we need to consider the time taken for acceleration.
To find the time t_acc taken for the motorcycle to catch up with the car after acceleration starts, we can use the formula:
Distance = Initial Velocity x Time + (1/2) x Acceleration x Time^2
57.0 m = 21.5 m/s x t_acc + (1/2) x 8.00 m/s^2 x t_acc^2
Simplifying the equation, we get a quadratic equation:
4t_acc^2 + 21.5t_acc - 57 = 0
Solving this equation using the quadratic formula, we find that t_acc = 2.498 s (approximately).
Now that we have t_acc, we can find the distance traveled by the motorcycle after t1 + t_acc:
Distance_motorcycle = Initial Velocity x Time + (1/2) x Acceleration x Time^2
Distance_motorcycle = 21.5 m/s x (3.00 s + 2.498 s) + (1/2) x 8.00 m/s^2 x (2.498 s)^2
Distance_motorcycle ≈ 160.3 m
Finally, we can find the time t2 when the motorcycle catches up with the car by equating the distances traveled by both:
Distance_car + Distance_motorcycle = 64.5 m + 160.3 m
Distance_car + Distance_motorcycle = 224.8 m
Now, the motorcycle catches up with the car when their distances are equal, so t2 is the time it takes for the motorcycle to travel 224.8 m.
To find t2, we can rearrange the formula for distance traveled:
Distance = Speed x Time
Time = Distance / Speed
t2 = Distance / Speed_motorcycle
t2 = 224.8 m / 21.5 m/s
t2 ≈ 10.5 s
Therefore, the motorcycle catches up with the car at approximately t2 = 10.5 seconds.