A 107.0 N grocery cart is pushed 14.0 m along an aisle by a shopper who exerts a constant horizontal force of 40.0 N. If all frictional forces are neglected and the cart starts from rest, what is the grocery cart's final speed?
The work done, which is
W = F*X = 40N*14.0m = ___ J ,
is equal to the increase in kinetic energy,
(1/2) M Vf^2 .
Therefore Vf = sqrt(2*f*X/M)
is the final speed.
To find the final speed of the grocery cart, we can use the work-energy principle. According to this principle, the work done on an object is equal to its change in kinetic energy.
First, let's determine the work done on the grocery cart. Work is defined as the force applied on an object multiplied by the distance over which the force is applied (W = F * d). In this case, the force applied is 40.0 N and the distance traveled is 14.0 m.
So, the work done on the grocery cart is W = 40.0 N * 14.0 m = 560 J (Joules).
Next, let's find the change in kinetic energy of the cart. The initial kinetic energy (K1) is zero because the cart starts from rest. The final kinetic energy (K2) is what we want to find.
Since work done is equal to the change in kinetic energy (W = ΔK), we can write:
ΔK = K2 - K1 = W
Substituting the values, we have:
K2 - 0 = 560 J
Therefore, the final kinetic energy of the grocery cart is 560 J.
Finally, let's convert the kinetic energy into speed. The formula for kinetic energy is K = 1/2 * m * v^2, where m is the mass of the cart and v is its speed.
Since we don't have the mass of the cart, we need to find a relationship between mass and weight.
Weight (W) is equal to mass (m) multiplied by the acceleration due to gravity (g). In this case, the weight is given as 107.0 N, so we can write:
W = m * g = 107.0 N
Now we can find the mass of the cart:
m = W / g = 107.0 N / 9.8 m/s^2 ≈ 10.92 kg
Substituting the values back into the kinetic energy formula:
560 J = 1/2 * 10.92 kg * v^2
Simplifying the equation, we can find the speed (v):
1120 J = 10.92 kg * v^2
v^2 = 1120 J / 10.92 kg
v^2 ≈ 102.56 m^2/s^2
Taking the square root of both sides:
v ≈ √(102.56 m^2/s^2)
v ≈ 10.13 m/s
Therefore, the grocery cart's final speed is approximately 10.13 m/s.