A car is speeding at constant velocity of 30ms-1 in an 80kmh-1 zone as it passes a stationary police motorcycle hiding. The motorcycle starts to accelerate at 5ms-2 at the same instant the car passes it and continues for 8 seconds. It then stops accelerating and maintains its speed until it catches the car.
a) How long does it take for the motorcycle to catch up to the car?
b) How far did the motorcycle travel before catching up to the car?
Please explain how you got the answer
To solve this problem, we need to break it down into several steps.
Step 1: Convert the car's speed from m/s to km/h:
The car's speed is given as 30 m/s. We need to convert it to km/h by multiplying it by 3.6 (since 1 m/s = 3.6 km/h).
Therefore, the car's speed is 30 m/s * 3.6 km/h/m/s = 108 km/h.
Step 2: Convert the acceleration of the motorcycle from m/s^2 to km/h^2:
The motorcycle's acceleration is given as 5 m/s^2. We need to convert it to km/h^2.
To do this, we multiply by (3.6)^2 since 1 m/s^2 = (3.6)^2 km/h^2.
Therefore, the motorcycle's acceleration is 5 m/s^2 * (3.6)^2 km/h^2 = 64.8 km/h^2.
Step 3: Calculate the time it takes for the motorcycle to catch up to the car:
We can use the formula: distance = initial velocity * time + (1/2) * acceleration * time^2.
Let's assume the time it takes for the motorcycle to catch up to the car is t seconds.
For the car:
distance = (108 km/h) * t
For the motorcycle:
distance = (0 km/h) * t + (1/2) * (64.8 km/h^2) * (t^2)
Note that the initial velocity is 0 km/h because the motorcycle starts from rest.
Setting the distances equal and solving for t, we get:
(108 km/h) * t = (1/2) * (64.8 km/h^2) * (t^2)
Simplifying the equation:
108 * t = 32.4 * (t^2)
32.4 * (t^2) - 108 * t = 0
32.4 * t^2 - 108 * t = 0
Factoring out t:
t * (32.4 * t - 108) = 0
Therefore, t = 0 or (32.4 * t - 108) = 0
Since time cannot be negative, we discard t = 0.
Solving for t, we get:
32.4 * t - 108 = 0
32.4 * t = 108
t = 108 / 32.4
t ≈ 3.33 seconds
Therefore, it takes approximately 3.33 seconds for the motorcycle to catch up to the car.
Step 4: Calculate the distance the motorcycle traveled before catching up to the car:
Using the distance formula for the motorcycle:
distance = (0 km/h) * t + (1/2) * (64.8 km/h^2) * (t^2)
Substituting the value of t we found:
distance = (0 km/h) * 3.33 seconds + (1/2) * (64.8 km/h^2) * (3.33 seconds)^2
Simplifying the equation:
distance = 0 + (1/2) * 64.8 km/h^2 * 11.0889 seconds^2
distance = 357.12 km
Therefore, the motorcycle traveled approximately 357.12 km before catching up to the car.
To find the answers to these questions, we need to understand the motion of both the car and the motorcycle. Let's break it down step by step:
Step 1: Convert the given velocities to meters per second (m/s).
The car is traveling at a constant velocity of 30 m/s.
The speed limit of 80 km/h is equivalent to 80,000 meters per hour or 22.22 meters per second (rounded to two decimal places).
Step 2: Determine the initial positions of the car and the motorcycle.
Since the motorcycle starts at rest, its initial position is the same as the car's when it passes the motorcycle.
Step 3: Calculate the time it takes for the motorcycle to catch up to the car.
To calculate the time, we need to consider the acceleration of the motorcycle. We'll use the equation:
distance = initial velocity * time + (1/2) * acceleration * (time^2)
We know that the initial velocity of the motorcycle is 0 m/s, and the acceleration is 5 m/s^2. The distance is the difference between the positions of the car and the motorcycle.
Substituting the values into the equation:
(distance between car and motorcycle) = 30t + (1/2) * 5 * t^2
(distance between car and motorcycle) = 30t + 2.5t^2
To find the time it takes for the motorcycle to catch up to the car, we need to solve this equation for t.
Step 4: Calculate the distance traveled by the motorcycle before catching up to the car.
Now, we can use the time obtained in the previous step to find the distance traveled by the motorcycle. We'll use the equation:
Distance = initial velocity * time + (1/2) * acceleration * (time^2)
In this case, the initial velocity of the motorcycle is 0 m/s, and the acceleration remains 5 m/s^2. The time is the same as the one calculated in step 3.
Substituting the values into the equation:
Distance = 0 * t + (1/2) * 5 * t^2
Distance = 2.5t^2
Let's plug the numbers and calculate the answers:
a) How long does it take for the motorcycle to catch up to the car?
Substituting the position difference equation from step 3:
30t + 2.5t^2 = 0
2.5t^2 + 30t = 0
t(2.5t + 30) = 0
This equation has two solutions: t = 0 (which is not relevant) and 2.4 seconds.
So, it takes approximately 2.4 seconds for the motorcycle to catch up to the car.
b) How far did the motorcycle travel before catching up to the car?
Using the distance equation from step 4:
Distance = 2.5 * (2.4)^2
Distance ≈ 14.4 meters
Therefore, the motorcycle travels approximately 14.4 meters before catching up to the car.