Question

A shopper in Whole Foods pushes their 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 (24kg), 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?

• 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?

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

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 force of 40 N at an angle of 30 degrees downward from the horizontal can be resolved into its horizontal and vertical components. The horizontal component of the force is:F_horizontal = F * cos(theta) = 40 N * cos(30) = 34.64 N

The work done by the shopper on the cart is equal to the force applied multiplied by the distance moved, multiplied by the cosine of the angle between the force and the displacement. In this case, the displacement is 15 m, and the angle between the force and the displacement is 30 degrees. Therefore:Work = F * d * cos(theta) = 34.64 N * 15 m * cos(30) = 448.5 J

Speed of the cart at the tofu section: The initial potential energy of the cart and shopper is equal to their combined mass (85 kg) multiplied by the acceleration due to gravity (9.81 m/s^2) multiplied by the height of Twin Peaks (600 m).

Therefore: Potential energy = m * g * h = 85 kg * 9.81 m/s^2 * 600 m = 498,690 J

Since the shopper neglects friction, all of the potential energy is converted into kinetic energy at the bottom of the hill. The kinetic energy of the cart and shopper can be found using the formula:

Kinetic energy = 0.5 * m * v^2

where m is the mass of the cart and shopper, and v is their speed.

Therefore:Kinetic energy = 0.5 * 85 kg * v^2 Since the potential and kinetic energies are equal, we can set them equal to each other and solve for the speed v: Potential energy = Kinetic energym * g * h = 0.5 * m * v^2v = sqrt(2 * g * h) = sqrt(2 * 9.81 m/s^2 * 600 m) = 109.43 m/s

Power exerted by the shopper in stopping the cart:The shopper brings the cart to rest over a distance of 15 m in a time of 2.7 s. The average force exerted by the shopper on the cart can be found using Newton's second law: F = m * a = m * (v_f - v_i) / twhere m is the mass of the cart, v_i is the initial speed of the cart (which is zero), v_f is the final speed of the cart, and t is the time taken to stop the cart. Solving for F gives:F = m * (v_f / t)The work done by the shopper in stopping the cart is equal to the force applied multiplied by the distance moved, which is 15 m. Therefore:Work = F * d = m * (v_f / t) * d = 24 kg * (0 - 0) / 2.7 s * 15 m = 0 JSince no work is done in bringing the cart to rest, the power exerted by the shopper is zero.Acceleration due to gravity on the unknown planet:The final speed of the 4 kg ball is 12 m/s, and it falls a distance of 60 m. The initial speed is zero, so the final velocity is equal to the velocity acquired due to gravity. The final velocity can be found using the formula:v_f^2 = v_i^2 + 2 * g * hwhere v_i is the initial velocity (which is zero), h is the height fallen, and g is the acceleration due to gravity.