At the local grocery store, you push a 14.0-kg shopping cart. You stop for a moment to add a bag of dog food to your cart. With a force of 12.0N, you now accelerate the cart from rest through a distance of 2.30m in 2.55s. What was the mass of the dog food?
Mc = 14 kg = Mass of cart
Md = Mass of dog food.
d = 0.5a*t^2 = 2.30 m.
0.5a*2.55^2 = 2.3
3.25a = 2.3
a = 0.707 m/s^2.
F = M*a
M = F/a
(Mc+Md) = F/a
F = 12 N.
a = 0.707 m/s^2
Solve for Md.
Well, aren't you just a strong shopper! Let's calculate the mass of the dog food, shall we?
First, we need to find the net force acting on the cart. We know that the force you applied to accelerate the cart is 12.0N. So, we need to subtract this force from the net force. The net force is given by Newton's second law, which states that Fnet = ma, where m is the mass of the cart and a is the acceleration.
Since the cart starts from rest, the initial velocity is zero. We can use the equation of motion to find the acceleration: s = ut + 0.5at^2, where s is the distance traveled, u is the initial velocity, a is the acceleration, and t is the time.
Substituting the given values, we have 2.30m = 0.5a(2.55s)^2. Solving for a, we get a = 2.30m / (0.5 * (2.55s)^2).
Now that we have the acceleration, we can use Fnet = ma to find the net force. The net force is the force of you pushing the cart minus the force of friction. But for fun, let's say the cart is on wheels made of slippery banana peels. That would mean the force of friction is negligible, so the net force is simply the force you applied.
Fnet = 12.0N = m * a
Now we can rearrange the equation to solve for mass:
m = Fnet / a = 12.0N / (2.30m / (0.5 * (2.55s)^2))
Now, let's plug in the numbers and crunch some math!
To find the mass of the dog food, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
1. First, let's calculate the acceleration of the shopping cart. We can use the kinematic equation:
Distance (d) = Initial velocity (v₀) * Time (t) + 0.5 * Acceleration (a) * Time squared (t²)
Rearranging the equation to solve for acceleration:
Acceleration (a) = 2 * (Distance (d) - Initial velocity (v₀) * Time (t)) / Time squared (t²)
Plugging in the given values:
Acceleration (a) = 2 * (2.30 m - 0 * 2.55 s) / (2.55 s)²
Acceleration (a) ≈ 1.04 m/s²
2. Now, we can determine the net force acting on the cart. Using Newton's second law:
Net Force (F) = Mass (m) * Acceleration (a)
Rearranging the equation to solve for mass:
Mass (m) = Net Force (F) / Acceleration (a)
Plugging in the given values:
Mass (m) = 12.0 N / 1.04 m/s²
Mass (m) ≈ 11.54 kg
Therefore, the mass of the dog food is approximately 11.54 kg.
To find the mass of the dog food, we can use Newton's second law of motion, which states that the force acting on an object is equal to the mass of the object multiplied by its acceleration.
First, let's calculate the acceleration of the shopping cart. We know that the initial velocity (v₀) is zero since it starts from rest, the distance (d) is 2.30m, and the time (t) is 2.55s.
We can use the formula:
acceleration (a) = (change in velocity) / time
Since the initial velocity is zero, the change in velocity is equal to the final velocity (v) we need to find.
Using the formula:
v = d / t
Substituting the given values, we get
v = 2.30m / 2.55s = 0.902 m/s
Now, using Newton's second law:
force (F) = mass (m) * acceleration (a)
Rearranging the formula, we can solve for the mass (m):
mass (m) = force (F) / acceleration (a)
Substituting the given values:
mass (m) = 12.0N / 0.902 m/s² = 13.30 kg
Therefore, the mass of the dog food is approximately 13.30 kg.