A 1500 kilogram car skids to a hold on a wet road where the coefficient of kinetic friction is 0.5. How fast was the car traveling if it leaves 65 meter long skid marks?
Well, that car must have been quite the "slippery" character! Let's calculate its speed by embracing our inner math clown.
We can use the formula for frictional force (Ff) to find the force holding back the car during the skid. Ff is equal to the coefficient of kinetic friction (μk) multiplied by the normal force (N). In this case, the normal force is equal to the weight of the car (mg).
So, Ff = μk * mg.
The force of friction can also be related to the acceleration (a) of the car using Newton's second law (F = ma). In this case, the force of friction would be equal to ma.
So, applying Newton's second law, we have ma = μk * mg.
We can cancel out the mass (m) from both sides of the equation, leaving us with a = μk * g.
The acceleration (a) can be related to the velocity (v) and the distance (d) using the equation v^2 = u^2 + 2ad, where u is the initial velocity (which we want to find) and v is the final velocity (which will be zero since the car comes to a halt).
Since the car skids to a stop, the final velocity (v) is 0. Plugging everything in, we get:
0 = u^2 + 2ad.
Rearranging the formula, we get:
u^2 = -2ad.
Now we can substitute the values:
u^2 = -2 * μk * g * d.
Simplifying this equation, we get:
u = sqrt(-2 * μk * g * d).
Now let's crunch some numbers!
With a coefficient of kinetic friction (μk) of 0.5, the acceleration due to gravity (g) being approximately 9.8 m/s^2, and a distance (d) of 65 meters, we can find the speed (u) by solving for u:
u = sqrt(-2 * 0.5 * 9.8 * 65).
Calculating that, we find that the car was traveling at approximately 31.6 m/s (or about 113.8 km/h) before it kissed those skid marks onto the road. Keep an eye on those "slippery" conditions next time!
To find the initial speed of the car, we can use the equation for kinetic friction:
frictional force = coefficient of kinetic friction × normal force
The normal force can be calculated using the equation:
normal force = mass × gravitational acceleration
The gravitational acceleration is approximately 9.8 m/s^2.
First, let's calculate the normal force:
normal force = 1500 kg × 9.8 m/s^2 = 14,700 N
Now, let's calculate the frictional force:
frictional force = coefficient of kinetic friction × normal force
= 0.5 × 14,700 N = 7,350 N
Using Newton's second law of motion, we can relate the net force to the acceleration:
net force = mass × acceleration
Since the car comes to a stop, the net force would be equal to the frictional force:
net force = frictional force = 7,350 N
We can rearrange the equation to solve for acceleration:
acceleration = net force / mass
acceleration = 7,350 N / 1500 kg ≈ 4.9 m/s^2
Now, let's use the equations of motion to find the initial speed of the car:
v^2 = u^2 + 2as
Where:
v = final velocity (0 m/s, since the car comes to a stop)
u = initial velocity (what we want to find)
a = acceleration (-4.9 m/s^2, since it is in the opposite direction of motion)
s = distance traveled (65 m)
Substituting the known values into the equation:
0 = u^2 + 2 × (-4.9 m/s^2) × 65 m
Simplifying the equation:
u^2 = 637 m^2/s^2
Taking the square root of both sides:
u = √637 m/s ≈ 25.3 m/s
Therefore, the car was traveling approximately 25.3 meters per second (m/s).
To determine the initial speed of the car, we can use the equation of motion that relates distance, acceleration, and initial velocity.
First, we need to find the acceleration which can be calculated using the coefficient of kinetic friction. The formula to calculate the frictional force is:
Frictional force = coefficient of kinetic friction * normal force
The normal force is the force exerted by the surface on the object and is equal to the weight of the car, which is the mass (1500 kg) multiplied by the acceleration due to gravity (9.8 m/s^2):
Normal force = mass * acceleration due to gravity
Now, we can calculate the frictional force:
Frictional force = coefficient of kinetic friction * normal force
Next, we can use Newton's second law of motion to relate the frictional force to acceleration:
Frictional force = mass * acceleration
Equating the two expressions for the frictional force and setting it equal to the product of mass and acceleration gives us:
coefficient of kinetic friction * normal force = mass * acceleration
Now we can substitute in the known quantities:
0.5 * (1500 kg)(9.8 m/s^2) = (1500 kg) * acceleration
Simplifying the equation gives:
(0.5)(1500 kg)(9.8 m/s^2) = 1500 kg * acceleration
Solving for acceleration gives:
acceleration = (0.5)(9.8 m/s^2)
acceleration = 4.9 m/s^2
Now, we can use the equation of motion to find the initial velocity. The equation is:
velocity^2 = initial velocity^2 + 2 * acceleration * distance
Since the car comes to a stop, the final velocity is 0, and the equation simplifies to:
0 = initial velocity^2 + 2 * acceleration * distance
Plugging in the values:
0 = initial velocity^2 + 2 * (4.9 m/s^2) * (65 m)
This equation can be solved for initial velocity:
initial velocity^2 = -2 * (4.9 m/s^2) * (65 m)
Taking the square root of both sides gives:
initial velocity = √[-2 * (4.9 m/s^2) * (65 m)]
Calculating the expression inside the square root:
initial velocity ≈ √[-637 m^2/s^2]
Since the velocity cannot be negative in this context, we can ignore the negative sign, and the final answer is:
initial velocity ≈ √637 m/s ≈ 25.23 m/s
Therefore, the car was traveling approximately 25.23 m/s when it skids to a stop.