A motorcycle traveling 95.0 km/hr approaches a car traveling in the same direction at 85.0 km/hr. When the motorcycle is 55.0 m behind the car, the rider accelerates and passes the car 18.0 s later. What is the acceleration of the motorcycle (in meters/second^2)?
Vo = 95000m/3600s = 26.4 m/s.
V = 85000m/3600s = 23.6 m/s.
Vo*t + 0.5a*t^2 = V*t + 55.
26.4*18 + 0.5a*18^2 = 23.6*18 + 55
a = ?
To calculate the acceleration of the motorcycle, we can use the equation of motion:
\[x = u*t + \frac{1}{2} * a * t^2\]
Where:
- x is the distance traveled
- u is the initial velocity
- t is the time taken
- a is the acceleration
Let's break down the problem into two parts:
1. When the motorcycle is behind the car:
The motorcycle is initially traveling at a constant velocity of 95.0 km/hr, and the car is traveling at a constant velocity of 85.0 km/hr. The relative velocity between the motorcycle and the car can be found by subtracting their velocities:
Relative Velocity = Motorcycle Velocity - Car Velocity
= 95.0 km/hr - 85.0 km/hr
= 10.0 km/hr
Since our equation of motion requires the velocities to be in meters per second, we need to convert km/hr to m/s:
1 km/hr = 1000 m/3600 sec ≈ 0.278 m/s
Therefore, the relative velocity is:
Relative Velocity = 10.0 km/hr * 0.278 m/s
= 2.78 m/s
The distance between the motorcycle and the car is 55.0 m. We can substitute the values into the equation of motion to find the time it takes for the motorcycle to catch up to the car:
55.0 m = 2.78 m/s * t + (1/2) * a * t^2
2. When the motorcycle accelerates and passes the car:
The motorcycle accelerates and passes the car 18.0 s later. We can now use the new velocity of the motorcycle to find the acceleration:
New velocity = Motorcycle velocity + (Acceleration * time)
The new velocity of the motorcycle can be calculated using the equation derived from the distance traveled:
New velocity = relative velocity + (acceleration * time)
Substituting the relative velocity and time:
New velocity = 2.78 m/s + (acceleration * 18.0 s)
Since the motorcycle passes the car, the distance traveled by the motorcycle will now be equal to the distance between the motorcycle and the car initially (55.0 m). We can substitute the values into the equation of motion:
55.0 m = New velocity * t + (1/2) * a * t^2
Now we have two equations and two unknowns (acceleration and time). By solving these equations simultaneously, we can find the acceleration of the motorcycle.