Question

A shopper in Whole Foods pushes their 24kg cart with a force of 40 N directed at an angle of 30 degrees downward form the horizontal. Find the work done by the shopper on the cart as he moves down a 15 m aisle to the tofu section.

If the mass of the cart, from problem 1, and we neglect friction, how fast will the cart be moving when it reaches the tofu section if it started from rest?


The shopper brings the cart to rest is 2.7 s when he reaches the tofu. What power does he exert in stopping the cart?




You take the bus with your bike to the top of Twin Peaks in San Francisco. You are facing a long descent: the top of Twin Peaks is 600 m higher than its base. The combined mass of you and your bicycle is 85 kg. How much gravitational potential
energy do you have before your descent?



You descend. If all that potential energy is converted to kinetic energy, what will your
speed be at the bottom?



On an unknown airless planet an astronaut drops a 4 kg ball from a 60 m ledge. The mass hits the bottom with a speed of 12 m/s. What is the acceleration of gravity g on
this planet?


The planet has a twin in an alternate universe with exactly the same acceleration of gravity. The difference is that this planet has an atmosphere. In this case, when dropped from a ledge with the same height, the 4 kg ball hits bottom at the speed of 9 m/s. How much energy is lost to air resistance during the fall?




A 1500 kg car starts at rest and speeds up to 3 m/s with a constant acceleration. If the car reaches its final speed in 1.2 s, what is its acceleration?

How far does the car travel in that time?



What is the car's gain in kinetic energy?


What power is exerted by the engine?

Answer 1

Work done by the shopper on the cart:

The component of the force in the direction of motion is Fcosθ = 40 N cos(30°) = 34.64 N. The work done is then:

W = Fcosθ x d = 34.64 N x 15 m = 519.6 J

Velocity of the cart:

The work done by the shopper is equal to the change in kinetic energy of the cart. The initial kinetic energy is zero, so:

(1/2)mv^2 = 519.6 J

where m is the mass of the cart and v is the velocity. Substituting m = 24 kg, we get:

v = sqrt((2 x 519.6 J) / 24 kg) = 7.98 m/s

Power exerted by the shopper:

The work done in bringing the cart to rest is also 519.6 J. The time taken is 2.7 s, so the power is:

P = W/t = 519.6 J / 2.7 s = 192.44 W

Gravitational potential energy:

The gravitational potential energy is given by:

U = mgh

where m is the combined mass of you and your bicycle, g is the acceleration due to gravity, and h is the height difference. Substituting m = 85 kg, g = 9.8 m/s^2, and h = 600 m, we get:

U = 85 kg x 9.8 m/s^2 x 600 m = 499800 J

Speed at the bottom of the descent:

The gravitational potential energy at the top is converted entirely to kinetic energy at the bottom:

(1/2)mv^2 = U

where m and v are the same as before. Substituting the values, we get:

v = sqrt((2 x 499800 J) / 85 kg) = 98.75 m/s

Acceleration due to gravity:

The final velocity and the height difference can be used to find the acceleration due to gravity:

v^2 = u^2 + 2gh

where u is the initial velocity, which is zero. Substituting the values, we get:

g = (v^2) / (2h) = (12 m/s)^2 / (2 x 60 m) = 4 m/s^2

Energy lost to air resistance:

The energy lost to air resistance is equal to the difference between the initial and final kinetic energies:

(1/2)mv_initial^2 - (1/2)mv_final^2

where m and v_final are the same as before, and v_initial is zero. Substituting the values, we get:

(1/2) x 4 kg x (0 m/s)^2 - (1/2) x 4 kg x (9 m/s)^2 = -162 J

The negative sign indicates that energy is lost, as expected.

Acceleration of the car:

The acceleration of the car is given by:

a = (v - u) / t

where u is the initial velocity, which is zero, v is the final velocity, and t is the time taken. Substituting the values, we get:

a = (3 m/s - 0 m/s) / 1.2 s = 2.5 m/s^2

Distance traveled by the car:

The distance traveled is given by:

d = ut + (1/2)at^2

where u and a are the same as before, and t is the time taken.