To solve the problem, we need to find the time it takes for Batman to attain the same speed as Robin.
Let's assume that it takes t seconds for Batman to attain Robin's speed.
We know that Batman is initially at rest while Robin is passing him at a speed of 60 km/h. Batman then starts accelerating at a rate of 10 km/h/s.
Using the equation of motion: final velocity (v) = initial velocity (u) + acceleration (a) * time (t)
The final velocity for Batman would be the same as Robin's speed, which is 60 km/h.
The initial velocity for Batman is 0 km/h.
The acceleration is 10 km/h/s.
The time taken is t seconds.
Therefore, the equation becomes: 60 km/h = 0 km/h + 10 km/h/s * t
Simplifying the equation: 60 km/h = 10 km/h/s * t
Dividing both sides of the equation by 10 km/h/s: t = 60 km/h / 10 km/h/s = 6 seconds
Hence, it takes Batman 6 seconds to attain the same speed as Robin.
Batman is sitting in the Batmobile at a stoplight. As the light turns green, Robin passes Batman in his lime-green Pinto at a constant speed of 60 km/ h. If Batman gives chase, accelerating at a constant rate of 10 km/ h/ s, determine a) how long it takes Batman to attain the same speed as Robin.
5 answers
b) how far Batman travels in this time.
To find how far Batman travels in this time, we need to use the equation of motion:
distance (d) = initial velocity (u) * time (t) + acceleration (a) * time (t)^2/2
In this case, the initial velocity for Batman is 0 km/h because he starts from rest.
The time taken is 6 seconds, as calculated in part a).
The acceleration is 10 km/h/s.
Plugging these values into the equation, we get:
distance (d) = 0 km/h * 6 s + 10 km/h/s * (6 s)^2 / 2
Simplifying the equation:
distance (d) = 0 + 10 km/h/s * (36 s^2) / 2
distance (d) = 10 km/h/s * 18 s^2
distance (d) = 180 km/s
Therefore, Batman travels a distance of 180 km in this time.
distance (d) = initial velocity (u) * time (t) + acceleration (a) * time (t)^2/2
In this case, the initial velocity for Batman is 0 km/h because he starts from rest.
The time taken is 6 seconds, as calculated in part a).
The acceleration is 10 km/h/s.
Plugging these values into the equation, we get:
distance (d) = 0 km/h * 6 s + 10 km/h/s * (6 s)^2 / 2
Simplifying the equation:
distance (d) = 0 + 10 km/h/s * (36 s^2) / 2
distance (d) = 10 km/h/s * 18 s^2
distance (d) = 180 km/s
Therefore, Batman travels a distance of 180 km in this time.
c) how long it takes for Batman to catch up to Robin.
To find how long it takes for Batman to catch up to Robin, we can set up a distance equation:
Distance traveled by Robin = Distance traveled by Batman
The distance traveled by Robin is given by the formula: distance = speed * time
Since Robin is traveling at a constant speed of 60 km/h, the distance traveled by Robin is 60t kilometers, where t is the time in hours.
The distance traveled by Batman can be found using the formula: distance = (initial velocity * time) + (1/2 * acceleration * time^2)
Since Batman starts from rest (initial velocity = 0 km/h), the distance traveled by Batman is given by 1/2 * acceleration * time^2 = (1/2 * 10 km/h/s) * t^2
Setting up the equation:
60t = (1/2 * 10 km/h/s) * t^2
Simplifying the equation:
60t = 5t^2
Rearranging the equation to quadratic form:
5t^2 - 60t = 0
Factoring out a common factor of t:
t(5t - 60) = 0
Setting each factor equal to zero:
t = 0 (This solution is not relevant in this context as it represents when Batman and Robin are at the same starting point.)
or
5t - 60 = 0
Solving for t:
5t = 60
t = 60/5
t = 12
Therefore, it takes Batman 12 hours to catch up to Robin.
Distance traveled by Robin = Distance traveled by Batman
The distance traveled by Robin is given by the formula: distance = speed * time
Since Robin is traveling at a constant speed of 60 km/h, the distance traveled by Robin is 60t kilometers, where t is the time in hours.
The distance traveled by Batman can be found using the formula: distance = (initial velocity * time) + (1/2 * acceleration * time^2)
Since Batman starts from rest (initial velocity = 0 km/h), the distance traveled by Batman is given by 1/2 * acceleration * time^2 = (1/2 * 10 km/h/s) * t^2
Setting up the equation:
60t = (1/2 * 10 km/h/s) * t^2
Simplifying the equation:
60t = 5t^2
Rearranging the equation to quadratic form:
5t^2 - 60t = 0
Factoring out a common factor of t:
t(5t - 60) = 0
Setting each factor equal to zero:
t = 0 (This solution is not relevant in this context as it represents when Batman and Robin are at the same starting point.)
or
5t - 60 = 0
Solving for t:
5t = 60
t = 60/5
t = 12
Therefore, it takes Batman 12 hours to catch up to Robin.