A 91.0 N grocery cart is pushed 17.1 m along an aisle by a shopper who exerts a constant horizontal force of 42.0 N. The acceleration of gravity is 9.81 m/s2 .
If all frictional forces are neglected and the cart starts from rest, what is the grocery cart’s final speed? Answer in units of m/s
m*g = 91 N.
m = 91/g = 91/9.81 = 9.29 kg.
a = Fap/m = 42/9.29 = 4.52 m/s^2.
V^2 = Vo^2 + 2a*d.
V^2 = 0 + 9.05*17.1 = 154.69
V = 12.44 m/s. = Final velocity.
To find the final speed of the grocery cart, we can use Newton's second law of motion and equations of motion.
1. Determine the net force acting on the cart:
Net force = applied force - force of gravity
= 42.0 N - 91.0 N
= -49.0 N (negative sign indicates opposite direction)
2. Calculate the acceleration of the cart:
Using Newton's second law of motion:
Net force = mass × acceleration
-49.0 N = 91.0 N × acceleration
acceleration = -49.0 N / 91.0 N
acceleration = -0.538 m/s^2 (negative sign indicates opposite direction)
3. Use the equation of motion to find the final speed:
vf^2 = vi^2 + 2 × acceleration × distance
Since the cart starts from rest (initial velocity, vi = 0), the equation simplifies to:
vf^2 = 2 × acceleration × distance
vf^2 = 2 × -0.538 m/s^2 × 17.1 m
vf^2 = -18.4 m^2/s^2 (since the acceleration is negative)
Taking the square root of both sides gives:
vf = √(-18.4)
Note: The square root of a negative number is not a real value, so the final speed is imaginary which implies that the cart does not reach the end of the aisle.
To determine the final speed of the grocery cart, we can use Newton's second law of motion, which states that the acceleration of an object is directly proportional to the net force applied to it and inversely proportional to its mass. Mathematically, this is represented by the equation:
F = m * a
Where F is the net force, m is the mass, and a is the acceleration.
In this case, the net force acting on the grocery cart is the horizontal force exerted by the shopper, which is 42.0 N. The mass of the cart, given as 91.0 N, can be converted to kg by dividing by the acceleration due to gravity (9.81 m/s^2):
m = 91.0 N / 9.81 m/s^2 ≈ 9.29 kg
Since the cart starts from rest, its initial velocity is 0 m/s. We can use the following equation to find the final velocity (v) of the cart:
v^2 = u^2 + 2as
Where u is the initial velocity (0 m/s), a is the acceleration, s is the distance traveled, and v is the final velocity.
We can solve for a using Newton's second law:
F = m * a
42.0 N = 9.29 kg * a
a = 42.0 N / 9.29 kg ≈ 4.52 m/s^2
Finally, we can substitute the values into the equation for final velocity (v):
v^2 = 0^2 + 2 * 4.52 m/s^2 * 17.1 m
v^2 = 2 * 4.52 m/s^2 * 17.1 m
v^2 = 154.68 m^2/s^2
To find v, we need to take the square root of both sides of the equation:
v = sqrt(154.68 m^2/s^2)
v ≈ 12.45 m/s
Therefore, the grocery cart’s final speed is approximately 12.45 m/s.