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 19.5 m/s , and the distance between them is 57.0 m . After t1 = 4.00 s , the motorcycle starts to accelerate at a rate of 7.00 m/s2 . The motorcycle catches up with the car at some time t2.How long does it take from the moment when the motorcycle starts to accelerate until it catches up with the car? In other words, find t2−t1.
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To find the time it takes for the motorcycle to catch up with the car, we need to determine the time it takes for the motorcycle to cover the initial distance between them while accelerating.
Here's how we can proceed:
Step 1: Calculate the distance covered by the car in time t1.
The distance covered by an object moving at a constant speed can be calculated using the formula: distance = speed × time.
The car's speed is given as 19.5 m/s, and the time t1 is given as 4.00 s. Therefore, the distance covered by the car in time t1 is:
distance_car = speed_car × t1 = 19.5 m/s × 4.00 s = 78.0 m
Step 2: Calculate the distance covered by the motorcycle while accelerating.
To calculate the distance covered by an object while accelerating, you can use the equation of motion: distance = initial velocity × time + (1/2) × acceleration × time².
Here, the initial velocity of the motorcycle is 19.5 m/s, the acceleration is 7.00 m/s², and the time is t2 - t1. However, we are trying to find t2 - t1, so let's call it Δt (change in time).
The distance covered by the motorcycle while accelerating can be calculated as:
distance_motorcycle = initial velocity_motorcycle × Δt + (1/2) × acceleration_motorcycle × Δt²
distance_motorcycle = 19.5 m/s × Δt + (1/2) × 7.00 m/s² × Δt²
Step 3: Equate the distances covered by the car and the motorcycle.
Since the motorcycle catches up with the car, the distances covered by both are the same. Therefore, we can equate the distance covered by the car in Step 1 with the distance covered by the motorcycle in Step 2.
78.0 m = 19.5 m/s × Δt + (1/2) × 7.00 m/s² × Δt²
This gives us a quadratic equation in terms of Δt. Let's simplify it.
Step 4: Solve the quadratic equation to find the value of Δt (t2 - t1).
Rearrange the equation in Step 3 to form a quadratic equation:
(1/2) × 7.00 m/s² × Δt² + 19.5 m/s × Δt - 78.0 m = 0
Now, you can solve this quadratic equation using the quadratic formula:
Δt = (-b ± √(b² - 4ac)) / (2a)
For our equation:
a = (1/2) × 7.00 m/s² = 3.5 m/s²
b = 19.5 m/s
c = -78.0 m
Plug in these values into the quadratic formula and solve for Δt. You'll obtain two possible solutions, one positive and one negative. Since we are interested in the time difference, t2 - t1, we only consider the positive solution.
Once you have the value of Δt (t2 - t1), you can calculate t2 by adding Δt to t1.
That's how you find the time it takes for the motorcycle to catch up with the car.