7. A cart like the one we use in lab accelerates down a 20° incline (and since the wheels are smooth, ignore friction for now). Break the force of gravity into components (leave mass as m for now), find the cart’s acceleration, and determine how long it should take the cart to get 75 cm down the track.

To solve this problem, we need to break down the force of gravity into components along the incline.

The force of gravity acting on the cart can be split into two components: one along the direction of the incline (mg sinθ) and one perpendicular to the incline (mg cosθ). Here, θ represents the angle of the incline.

To find the acceleration of the cart, we need to consider the force contributing to its motion, which is the component of gravity along the incline (mg sinθ). We can use Newton's second law, which states that the net force on an object is equal to its mass (m) multiplied by its acceleration (a). In this case, the net force is the force along the incline, so:

mg sinθ = ma

We can rearrange this equation to solve for the acceleration:

a = (mg sinθ) / m

The mass cancels out, giving us:

a = g sinθ

Where g is the acceleration due to gravity (approximately 9.8 m/s²). Now we have the value for acceleration.

To determine the time it takes for the cart to travel a distance of 75 cm down the track, we need to use the equation of motion:

s = ut + (1/2)at²

Where:
s = distance (75 cm, which is equivalent to 0.75 m)
u = initial velocity (assuming the cart starts from rest, u = 0)
a = acceleration (from the previous calculation)
t = time (the value we want to find)

Rearranging the equation:

t = √(2s / a)

Now, we can substitute the values we have into the equation to find the time it takes for the cart to travel 75 cm down the track.

To solve this problem, we need to break the force of gravity into components and use them to find the acceleration of the cart.

1. Break the force of gravity into components:
The force of gravity acting on the cart can be broken into two components: the force parallel to the incline and the force perpendicular to the incline.

The force parallel to the incline can be calculated using:

F_parallel = m * g * sin(theta)

The force perpendicular to the incline is:

F_perpendicular = m * g * cos(theta)

Where:
m is the mass of the cart
g is the acceleration due to gravity (approximately 9.8 m/s^2)
theta is the angle of the incline (20°)

2. Calculate the acceleration of the cart:
The net force acting on the cart is equal to the force parallel to the incline. Since there is no friction, this force will accelerate the cart down the incline.

Net force = F_parallel

Since F_parallel = m * g * sin(theta), the acceleration can be calculated using Newton's second law:

a = Net force / m
a = (m * g * sin(theta)) / m
a = g * sin(theta)

3. Determine how long it should take the cart to get 75 cm down the track:
We can use the equations of motion to find the time it takes for the cart to travel a certain distance.

The equation for the displacement of an object with constant acceleration is:

s = ut + (1/2) * a * t^2

Where:
s is the distance traveled
u is the initial velocity (assumed to be 0 in this case)
a is the acceleration
t is the time

In this case, we want to find the time, so we rearrange the equation:

t = sqrt((2 * s) / a)

Using the given distance of 75 cm, we can calculate the time:

t = sqrt((2 * 75 cm) / (g * sin(theta)))

Note: Make sure to convert the distance to meters before plugging it into the equation.

Now you can substitute the values into the formulas and calculate the acceleration and time accordingly.