A car with a mass of 1100kg can accelerate on a level road from rest to 2m/s in 14sec.what is the steepest slope this car can climb.(hint:the plane is smooth.)
the acceleration is
... 2 m/s / 14 s = .14 m/s²
as the car goes up a slope, the gravitational acceleration on the car is proportional to the sine of the angle of the slope
sin(Θ) = .14 m/s² / 9.8 m/s² = .014
Θ = 0.8º
Well, let's calculate the slope that this car can conquer! But before we dive into the calculations, let me tell you a fun fact. Did you know that cars love slopes? They always gravitate towards them! Anyway, let's get back to the question at hand.
To solve this, we need to use Newton's second law of motion, which states that force equals mass times acceleration (F = ma). In this case, the car's acceleration is 2 m/s^2 (since it reaches 2 m/s in 14 seconds), and its mass is given as 1100 kg.
Now, let's consider the forces acting on the car. When climbing a slope, there are two main forces: gravity and the force due to the slope. The force due to gravity is given by the equation Fg = mg, where m is the mass of the car, and g is the acceleration due to gravity (approximately 9.8 m/s^2). The force due to the slope is given by Fs = m * a_s, where a_s is the acceleration due to the slope.
Since the car is moving uphill, the force due to the slope should oppose gravity. Therefore, we can express the force due to the slope as Fs = -Fg.
Now, let's put it all together. We have Fs = -Fg = m * a_s. Rearranging the equation, we get a_s = -Fg / m.
Since we want to find the steepest slope the car can climb, we need to find the maximum value of a_s. The steepest slope will be when a_s reaches its maximum value, which occurs when a_s = -g.
Plugging in the values, we have a_s = -9.8 m/s^2. Therefore, the steepest slope this car can climb is equal to -9.8 m/s^2.
So, it seems our car is a bit of a downhill enthusiast since it can only handle slopes with a maximum angle of -9.8 m/s^2. Better stock up on some brake pads!
To determine the steepest slope the car can climb, we need to calculate the maximum angle of inclination the car can handle.
Step 1: Calculate the force required to accelerate the car from rest to 2 m/s.
Using Newton's second law of motion (F = m * a), where F is the force, m is the mass, and a is the acceleration:
F = m * a
F = 1100 kg * (2 m/s^2)
F = 2200 N
Step 2: Calculate the force due to gravity acting against the car on a slope.
The force due to gravity can be calculated using the equation Fg = m * g, where Fg is the force due to gravity, m is the mass, and g is the acceleration due to gravity (approximately 9.8 m/s^2):
Fg = m * g
Fg = 1100 kg * 9.8 m/s^2
Fg = 10780 N
Step 3: Calculate the maximum force the car can handle on a slope.
The maximum force the car can handle is the difference between the force required to accelerate and the force due to gravity acting against the car:
Fmax = F - Fg
Fmax = 2200 N - 10780 N
Fmax = -8580 N
(Note: The negative sign indicates that the force is acting against the car's motion.)
Step 4: Calculate the angle of inclination using the formula tan(theta) = Fmax / Fg.
theta = arctan(Fmax / Fg)
theta = arctan(-8580 N / 10780 N)
theta ≈ -38.3 degrees
Step 5: Calculate the steepest slope the car can climb.
The steepest slope the car can climb is the incline with an angle of inclination equal to the calculated maximum angle. However, since the angle is negative, we consider its absolute value:
Steepest slope = |theta|
Steepest slope ≈ 38.3 degrees
Therefore, the steepest slope this car can climb on a smooth plane is approximately 38.3 degrees.
To determine the steepest slope the car can climb, we need to consider the forces acting on the car and use Newton's second law of motion.
First, let's calculate the net force acting on the car when it accelerates from rest to 2 m/s in 14 seconds. We can use the equation:
Net Force = Mass x Acceleration
Given:
Mass (m) = 1100 kg
Final velocity (v) = 2 m/s
Time taken (t) = 14 sec
Acceleration (a) can be calculated using the formula:
Acceleration = (Final Velocity - Initial Velocity) / Time
Acceleration = (2 m/s - 0 m/s) / 14 sec
Acceleration = 2 m/s / 14 sec
Acceleration = 0.14 m/s²
Substituting the values, we have:
Net Force = 1100 kg x 0.14 m/s²
Net Force = 154 N
Next, let's consider the forces acting on the car on the steepest slope it can climb. Since the plane is smooth (frictionless), the only forces acting on the car are the gravitational force and the normal force.
The gravitational force is given by:
Gravitational Force = Mass x Gravitational Acceleration
Gravitational Acceleration (g) on Earth is approximately 9.8 m/s².
Gravitational Force = 1100 kg x 9.8 m/s²
Gravitational Force = 10780 N
The normal force is the force exerted by the surface perpendicular to the car. On a level road, the normal force is equal in magnitude and opposite in direction to the gravitational force. Therefore, the magnitude of the normal force is also 10780 N.
For the car to climb the steepest slope without sliding back, the net force acting on the car must be equal to the difference between the gravitational force and the normal force. That is:
Net Force = Gravitational Force - Normal Force
154 N = 10780 N - Normal Force
To calculate the normal force, rearrange the equation:
Normal Force = 10780 N - 154 N
Normal Force = 10626 N
Now, we need to find the angle of the steepest slope the car can climb. The angle θ can be calculated using the equation:
sin θ = Normal Force / Gravitational Force
sin θ = 10626 N / 10780 N
Now, find the inverse sine (sine^(-1)) to get the angle θ:
θ = sin^(-1) (10626 N / 10780 N)
Using a calculator, we find θ to be approximately 67.5 degrees.
Therefore, the car can climb a slope with a maximum angle of approximately 67.5 degrees.