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 19.0m/s , and the distance between them is 52.0m . After t1 = 5.00s , the motorcycle starts to accelerate at a rate of 5.00m/s2. 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

Question

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 19.0m/s , and the distance between them is 52.0m . After t1 = 5.00s , the motorcycle starts to accelerate at a rate of 5.00m/s2. 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

Answer 1

Let t=0 be the time starting when the bike starts to accelerate. Since they were both going at the same speed, you just want to know how long it takes the bike to travel the extra 52m.

2.5t^2 = 52
t = 4.56 s

Answer 2

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Answer 3

To find the time it takes for the motorcycle to catch up with the car, we need to first determine the distance traveled by the car during the time interval t2 - t1, where t2 is the time it takes for the motorcycle to catch up.

Let's break down the problem:

1. Initial conditions:
- Car speed (v_car) = 19.0 m/s
- Motorcycle speed (v_motorcycle) = 19.0 m/s
- Initial distance between them (d_initial) = 52.0 m

2. Time interval after the motorcycle starts to accelerate (t2 - t1):
- Acceleration of the motorcycle (a_motorcycle) = 5.00 m/s²
- Acceleration time (t2 - t1) = ?

3. Calculate the distance traveled by the car during the time interval t2 - t1:
- Initial velocity of the car (v_initial) = 19.0 m/s
- Time interval (t2 - t1) = ?
- Distance traveled by the car (d_car) = ?

4. Equate the distances traveled by the car and the motorcycle:
- Distance traveled by the motorcycle (d_motorcycle) = d_car

Now let's solve the problem step by step:

Step 1: Calculate the distance traveled by the car during the time interval t2 - t1.

To find the distance traveled by an object when it is moving with constant velocity, we can use the formula:

distance = initial velocity * time

Here, the initial velocity of the car (v_initial) is given as 19.0 m/s, and the time interval (t2 - t1) is unknown. So, the distance traveled by the car (d_car) is:

d_car = v_initial * (t2 - t1)

Step 2: Equate the distances traveled by the car and the motorcycle:

Since the motorcycle catches up with the car, the distance traveled by the motorcycle will be equal to the distance traveled by the car:

d_motorcycle = d_car

Step 3: Solve for the time interval (t2 - t1):

Using the equation from Step 2 and substituting the formula for distance traveled by the car from Step 1, we can solve for (t2 - t1):

d_motorcycle = d_car
v_motorcycle * (t2 - t1) = v_initial * (t2 - t1)
19.0 * (t2 - t1) = 19.0 * (t2 - t1)

The time interval (t2 - t1) cancels out on both sides of the equation.

Therefore, the time interval (t2 - t1) can be any value since it doesn't affect the equation. The motorcycle will catch up with the car immediately after it starts to accelerate.

Thus, t2 - t1 = 0 s.