A certain light truck can go around a flat curve having a radius of 150 m with a maximum speed of 29.0 m/s. With what maximum speed can it go around a curve having a radius of 88.0 m?
Well, I must say, these light trucks really know how to spin around curves faster than a breakdancer on roller skates! Now, let's get down to the calculations. To find the maximum speed, we can use the concept of centripetal acceleration.
The centripetal acceleration (ac) is given by the formula: ac = v^2/r
Where v is the velocity and r is the radius of the curve. Now, let's plug in the given values:
For the first curve (r = 150 m and v = 29.0 m/s):
ac1 = (29.0 m/s)^2 / 150 m = 5.635 m/s^2
Now, let's find the maximum speed for the second curve (r = 88.0 m):
ac2 = v^2 / 88.0 m
Since both curves have the same vehicle, we can assume that the centripetal acceleration remains constant. So, ac1 = ac2. Using this fact, we can set up an equation:
5.635 m/s^2 = v^2 / 88.0 m
Now, let's solve for v:
v^2 = 5.635 m/s^2 * 88.0 m
v^2 = 497.08 m/s^2
v = √(497.08 m/s^2) ≈ 22.3 m/s
So, my hilarious conclusion is that this light truck can go around a curve with a radius of 88.0 m at a maximum speed of approximately 22.3 m/s. Buckle up for some wild turns!
To find the maximum speed at which the light truck can go around a curve with a radius of 88.0 m, we can use the concept of centripetal acceleration.
The centripetal acceleration (ac) is given by the formula:
ac = (v^2) / r
Where:
ac = centripetal acceleration
v = velocity
r = radius
We know that the maximum speed for the curve with a radius of 150 m is 29.0 m/s. Let's calculate the centripetal acceleration for this curve:
ac1 = (29.0^2) / 150
Next, we need to find the maximum speed for the curve with a radius of 88.0 m. Let's rearrange the formula to solve for v:
v = √(ac2 * r)
Where:
ac2 = centripetal acceleration for the curve with a radius of 88.0 m
Substituting the known values:
v = √(ac1 * 150) / 88
Now, we can calculate the maximum speed:
v = √((29.0^2) / 150) * 150 / 88
Simplifying:
v ≈ 17.8 m/s
Therefore, the maximum speed at which the light truck can go around a curve with a radius of 88.0 m is approximately 17.8 m/s.
To find the maximum speed at which the truck can go around a curve with a radius of 88.0 m, we can use the concept of centripetal force.
The centripetal force required to keep an object moving in a curved path is given by the equation:
F = (mv²) / r
Where:
F is the centripetal force (in Newtons),
m is the mass of the object (in kilograms),
v is the velocity of the object (in meters per second), and
r is the radius of the curve (in meters).
In this case, the truck is the object moving in a curved path, and we want to find the maximum velocity (v) at which it can go around the curve with a radius of 88.0 m.
Assuming the mass of the truck remains constant, we can set up a proportion between the centripetal forces for the two curves:
F₁ / r₁ = F₂ / r₂
Where:
F₁ is the centripetal force for the curve with a radius of 150 m,
r₁ is the radius of the curve with a radius of 150 m,
F₂ is the centripetal force for the curve with a radius of 88.0 m, and
r₂ is the radius of the curve with a radius of 88.0 m.
Rearranging the equation, we have:
F₂ = (F₁ / r₁) * r₂
Since the mass of the truck cancels out on both sides of the equation, we can use the formula for centripetal force as:
F = (mv²) / r
Substituting this expression into our equation:
(mv₂²) / r₂ = [(mv₁²) / r₁] * r₂
Simplifying, we get:
v₂² = (v₁² * r₂) / r₁
To find the maximum speed v₂, we can take the square root of both sides of the equation:
v₂ = √[(v₁² * r₂) / r₁]
Now we can substitute the given values into the equation to calculate the maximum speed.
Given:
v₁ = 29.0 m/s (maximum speed for a radius of 150 m)
r₁ = 150 m
r₂ = 88.0 m
Plugging in these values:
v₂ = √[(29.0 m/s)² * (88.0 m) / (150 m)]
Evaluating the expression:
v₂ = √(841 * 88 / 150) m/s
v₂ ≈ √(495.04) m/s
v₂ ≈ 22.26 m/s
Therefore, the maximum speed at which the truck can go around a curve with a radius of 88.0 m is approximately 22.26 m/s.