A car coasts (engine off) up a 30 degree grade. If the speed of the car is 25m/s at the bottom of the grade, what is the distance traveled by the car before it comes to rest?
m•a=m•g•sinα
a=g•sinα
s=v²/2•a=
= v²/2•g•sinα=
=25²/2•9.8•0.5=
=63.78 m
Where did the Sin(A) come from
Why did the car decide to coast up a 30-degree grade? Because it wanted to show off its uphill skills without the help of its engine, of course! Now, let's calculate the distance it traveled before coming to a stop.
To solve this problem, we need to consider the forces acting on the car. When it's on the incline, there are two main forces at play: the force of gravity pulling it downhill and the normal force pushing it perpendicularly to the incline. Since the car is coasting, there is no engine force.
With these forces in mind, we can use some good-old trigonometry. The component of gravity acting parallel to the incline is mg*sin(30) (where m is the mass of the car and g is the acceleration due to gravity). We can equate this force to the friction force opposing the car's motion (since it's coasting), which is μmg*cos(30) (where μ is the coefficient of friction). Since we're interested in the distance traveled, we can use the work-energy theorem.
The work done by the friction force is given by the equation W = ΔKE, where ΔKE is the change in kinetic energy. The initial kinetic energy (KE) is 0.5mv² (where v is the initial velocity), and the final kinetic energy is 0 since the car comes to a stop.
Setting up the equation, we have:
0.5mv² = μmg*cos(30)d,
where d is the distance traveled before coming to a stop.
Now, let's plug in the values:
0.5*(m)*(25^2) = μ*(m)*9.8*cos(30)*d.
Now, we can cancel out the mass of the car (m) on both sides and simplify the equation:
0.5*25^2 = 9.8*μ*cos(30)*d.
Simplifying further:
(0.5)*(625) = (9.8)*(0.866)*(d),
312.5 = 8.5d.
Finally, dividing both sides by 8.5 to solve for d:
d = 36.76470588235294.
So, the car traveled approximately 36.76 meters before it gracefully came to a stop. I hope that put a smile on your face!
To find the distance traveled by the car before it comes to rest, we can use the concept of energy conservation.
Step 1: Determine the change in potential energy.
The change in potential energy can be calculated using the formula:
ΔPE = m * g * h,
where ΔPE is the change in potential energy, m is the mass of the car, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the change in height.
Since the car starts at the bottom and comes to rest at the top of the grade, the change in height is the vertical distance traveled up the grade. This can be calculated using trigonometry:
h = d * sin(θ),
where h is the vertical distance, d is the distance traveled along the grade, and θ is the angle of the grade (30 degrees).
Step 2: Calculate the change in potential energy.
Substituting the value of h into the formula, we have:
ΔPE = m * g * (d * sin(θ)).
Step 3: Calculate the initial kinetic energy.
The initial kinetic energy of the car can be calculated using the formula:
KE = 0.5 * m * v²,
where KE is the kinetic energy, m is the mass of the car, and v is the initial speed of the car.
Step 4: Apply the principle of energy conservation.
According to the principle of conservation of energy, the change in potential energy is equal to the initial kinetic energy:
ΔPE = KE.
Substituting the formulas for ΔPE and KE, we have:
m * g * (d * sin(θ)) = 0.5 * m * v².
Step 5: Solve for the distance traveled.
Canceling out the mass, we get:
g * (d * sin(θ)) = 0.5 * v².
Simplifying further, we have:
d * sin(θ) = (0.5 * v²) / g.
Finally, solving for d, the distance traveled by the car before it comes to rest, we have:
d = (0.5 * v²) / (g * sin(θ)).
Substituting the given values (v = 25 m/s and θ = 30 degrees), we can calculate the distance traveled by the car.
To find the distance traveled by the car before it comes to rest, we need to calculate the work done against gravity.
First, we need to determine the vertical component of the car's velocity. We can do this by multiplying the car's initial velocity (25 m/s) by the sine of the angle of the grade (30 degrees).
Vertical component of velocity = 25 m/s * sin(30 degrees)
Next, we need to find the time it takes for the car to come to rest. Since the car is coasting and the engine is off, the only force opposing its motion is due to gravity. This force is equal to the weight of the car, which can be calculated by multiplying the mass of the car by the acceleration due to gravity (9.8 m/s^2).
Weight = mass * acceleration due to gravity
To find the time, we divide the vertical component of velocity by the acceleration due to gravity:
Time = Vertical component of velocity / acceleration due to gravity
Finally, we can calculate the distance traveled using the formula:
Distance = Initial velocity * Time - (1/2) * acceleration due to gravity * Time^2
Plugging in the values we've calculated:
Distance = 25 m/s * cos(30 degrees) * Time - (1/2) * 9.8 m/s^2 * Time^2
Simplifying further will give us the distance traveled by the car before it comes to rest.