A 85.0 N grocery cart is pushed 13.0 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 mass of the cart is M = W/g = 8.67 kg
The cart acquires kinetic energy equal to the work done on it.
(M/2) V^2 = F * X = 40 * 13 joules
Use mass M and the formula above to calculate V.
To find the grocery cart's final speed, we can use the formula for the work done on an object. The work done on an object is equal to the change in its kinetic energy. Since the cart starts from rest, its initial kinetic energy is zero.
The work done on the cart can be calculated as the product of the force applied and the distance moved.
Work (W) = Force (F) × Distance (d)
W = 40.0 N × 13.0 m
W = 520.0 N·m
Since work is equal to the change in kinetic energy, we can equate it to the final kinetic energy (KE):
KE = W
Since the initial kinetic energy is zero, the final kinetic energy is equal to the work done on the cart.
KE = 520.0 N·m
The formula for kinetic energy is given by:
KE = (1/2) × mass × velocity^2
We are given the weight (W) of the cart, but we'll need to find the mass (m) of the cart using the formula:
Weight (W) = mass (m) × gravity (g)
The weight of the cart is given as 85.0 N. The acceleration due to gravity is approximately 9.8 m/s^2.
85.0 N = m × 9.8 m/s^2
m = 85.0 N / 9.8 m/s^2
m = 8.67 kg (rounded to two decimal places)
Now we can solve for the final velocity (v).
520.0 N·m = (1/2) × 8.67 kg × v^2
Simplifying the equation:
520.0 N·m = 4.335 kg × v^2
Dividing both sides by 4.335 kg:
v^2 = 120.0 m^2/s^2
Taking the square root of both sides to find v:
v = √120.0 m^2/s^2
v ≈ 10.95 m/s
Therefore, the grocery cart's final speed is approximately 10.95 m/s.
To find the grocery cart's final speed, we need to apply Newton's second law of motion and use the equation relating force, mass, and acceleration.
The equation is:
F = m * a
Where F is the net force acting on the object, m is the mass of the object, and a is the acceleration.
In this case, the net force is the horizontal force applied by the shopper, and the mass is the mass of the grocery cart. The acceleration can be calculated using the kinematic equation:
v^2 = u^2 + 2 * a * s
Where v is the final velocity, u is the initial velocity (which is zero since the cart starts from rest), a is the acceleration, and s is the distance traveled.
Rearranging the equation, we get:
a = (v^2 - u^2) / (2 * s)
Now, we can substitute the values into the equations. The force exerted by the shopper is 40.0 N, and the mass of the cart is 85.0 N. The distance traveled is 13.0 m, and the initial velocity is zero.
Using the equation F = m * a, we can solve for the acceleration:
40.0 N = 85.0 N * a
a = 40.0 N / 85.0 N
a ≈ 0.471 m/s^2
Now, substituting the values into the kinematic equation:
v^2 = 0^2 + 2 * 0.471 m/s^2 * 13.0 m
v^2 = 0 + 9.03 m^2/s^2
v ≈ √9.03 m^2/s^2
v ≈ 3.00 m/s
Therefore, the grocery cart's final speed is approximately 3.00 m/s.