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.
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.
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.
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.
To determine how long it takes Batman to attain the same speed as Robin, we need to find the time it takes for Batman to match Robin's speed.
Let's denote:
- Batman's initial speed as v0 (at the moment the light turns green)
- Robin's constant speed as R
- Batman's constant acceleration as a
Given:
Robin's speed (R) = 60 km/h
Batman's acceleration (a) = 10 km/h/s
First, let's convert Robin's speed from km/h to m/s:
Speed in m/s = (Speed in km/h) × (1 km/3.6 m)
Robin's speed (R) = 60 km/h = 60 × (1 km/3.6 m) = 16.67 m/s
Since Robin's speed is constant, let's find the time it takes for Batman to match that speed.
Using the equation of motion:
Final speed (v) = Initial speed (v0) + (acceleration (a) × time (t))
We want Batman's final speed (v) to be equal to Robin's speed (R). Thus:
v = R = 16.67 m/s
v0 = 0 (since Batman starts from rest)
The equation becomes:
R = 0 + (a × t)
Now, let's solve for time (t):
t = R / a
Substituting the known values:
t = 16.67 m/s / 10 km/h/s × (1 km/3.6 m) = 0.46 seconds
Therefore, it will take Batman approximately 0.46 seconds to match Robin's speed.
To determine how long it takes for Batman to attain the same speed as Robin, we need to calculate the time it takes for Batman to accelerate to Robin's speed.
First, let's convert Robin's speed from kilometers per hour to meters per second, which will provide consistent units for the calculation. We know that 1 kilometer is equal to 1000 meters, and 1 hour is equal to 3600 seconds. Therefore, Robin's speed of 60 km/h can be converted to (60,000 m)/(3600 s), which simplifies to 16.67 m/s.
Next, we need to determine Batman's acceleration. Batman's acceleration is given as a constant rate of 10 km/h/s. We need to convert this to meters per second squared (m/s²) to maintain consistent units. Similar to the previous conversion, we know 1 hour is equal to 3600 seconds. Therefore, Batman's acceleration is (10,000 m)/(3600 s²), which simplifies to 2.78 m/s².
To find the time it takes for Batman to attain Robin's speed, we can use the following kinematic equation:
v = u + at
where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time.
Since we want Batman's final velocity to be equal to Robin's velocity, we can set v = 16.67 m/s.
However, Batman's initial velocity is 0 (standing still at the stoplight), so u = 0 m/s.
Using the equation v = u + at and rearranging it to solve for time (t), we get:
t = (v - u) / a
Substituting in the values, we have:
t = (16.67 m/s - 0 m/s) / 2.78 m/s²
Simplifying the equation, we find:
t = 6 seconds
Therefore, it will take Batman approximately 6 seconds to achieve the same speed as Robin.