The mass of a sports car is 1200 kg. The shape of the car is such that the aerodynamic drag coefficient is 0.250 and the frontal area is 2.30 m2. Neglecting all other sources of friction, calculate the initial acceleration of the car ,if it has been traveling at 85 km/h and is now shifted into neutral and is allowed to coast. (Take the density of air to be 1.295 kg/m2.)

To calculate the initial acceleration of the car, we first need to calculate the aerodynamic drag force acting on it. The formula for drag force is given by:

F_drag = 0.5 * ρ * C_d * A * v^2

where:
F_drag = aerodynamic drag force (unknown)
ρ = air density = 1.295 kg/m²
C_d = drag coefficient = 0.250
A = frontal area = 2.30 m²
v = velocity in m/s (we'll calculate this from the given 85 km/h)

First, let's convert the car's speed to meters per second:

85 km/h * (1000 m/km) * (1 h/3600 s) = 85 * (10/36) = 850/36 ≈ 23.61 m/s

Now, we can calculate the drag force:

F_drag = 0.5 * 1.295 * 0.250 * 2.30 * (23.61)^2 = 0.5 * 1.295 * 0.250 * 2.30 * (556.4921) ≈ 89.04 N

Now that we have the drag force, we can calculate the acceleration by using Newton's 2nd Law of Motion:

F = m * a

Where:
F = force (89.04 N in this case, acting on the car)
m = mass (1200 kg)
a = acceleration (unknown)

Since the car is in neutral, the only force acting on it is the drag force, which opposes its motion. Therefore:

a = F_drag / m = 89.04 N / 1200 kg ≈ 0.0742 m/s²

However, the acceleration is negative since it opposes the direction of motion:

a = -0.0742 m/s²

So, the initial acceleration of the car is approximately -0.0742 m/s².

To find the initial acceleration of the car, we can start by calculating the air resistance acting on it. Air resistance can be determined using the equation:

F_drag = 0.5 * Cd * A * ρ * v^2

where:
F_drag is the air resistance force,
Cd is the drag coefficient,
A is the frontal area of the car,
ρ is the density of air, and
v is the velocity of the car.

First, let's convert the velocity from km/h to m/s:
v = 85 km/h * (1000 m / 1 km) * (1 h / 3600 s) = 23.61 m/s

Now, we can substitute the given values into the equation to find the air resistance force:
F_drag = 0.5 * 0.250 * 2.30 m^2 * 1.295 kg/m^3 * (23.61 m/s)^2

Calculating this expression:
F_drag ≈ 0.5 * 0.250 * 2.30 * 1.295 * 558.32

F_drag ≈ 113.423 N

The net force acting on the car is equal to the force of air resistance since all other frictional sources are neglected.

Using Newton's second law of motion, we can relate the net force with the acceleration of the car:

F_net = m * a

where:
F_net is the net force acting on the car,
m is the mass of the car, and
a is the acceleration of the car.

Rearranging the equation, we can solve for acceleration:
a = F_net / m = F_drag / m

Substituting the values:
a = 113.423 N / 1200 kg

Calculating this expression:
a ≈ 0.0945 m/s^2

Therefore, the initial acceleration of the car, when it is shifted into neutral and allowed to coast, is approximately 0.0945 m/s^2.

To calculate the initial acceleration of the car when it is shifted into neutral and allowed to coast, we can use the equation of motion:

F_net = m * a

Where:
F_net is the net force acting on the car,
m is the mass of the car, and
a is the acceleration.

To calculate the net force, we need to consider the forces acting on the car:

1. Aerodynamic drag force (Fd):
Fd = (1/2) * Cd * A * ρ * v^2
where Cd is the drag coefficient,
A is the frontal area of the car,
ρ is the density of air, and
v is the velocity of the car.

2. Net force (F_net):
F_net = - Fd
Since the car is allowed to coast, there is no external force applied, and the net force is equal to the drag force acting in the opposite direction.

Now, let's calculate the initial acceleration:

Step 1: Convert the velocity from km/h to m/s.
v = 85 km/h = (85 * 1000) / (60 * 60) m/s = 23.61 m/s

Step 2: Plug in the given values into the aerodynamic drag force equation.
Cd = 0.250
A = 2.30 m^2
ρ = 1.295 kg/m^3
v = 23.61 m/s

Fd = (1/2) * 0.250 * 2.30 * 1.295 * (23.61^2)

Step 3: Calculate the net force.
F_net = - Fd

Step 4: Calculate the acceleration using Newton's second law.
a = F_net / m

Plug in the values of F_net and m into the equation to get the acceleration.

Note: The negative sign indicates that the acceleration is in the opposite direction of the initial velocity.

By following these steps, you can calculate the initial acceleration of the car.