A block of mass $m$, after sliding down a frictionless incline, strikes another block of mass $M$ that is attached to a spring of spring constant $k$ (see below). The blocks stick together upon impact and travel together. (a) Find the compression of the spring in terms of $m, M, h, g$, and $k$ when the combination comes to rest. Hint: The speed of the combined blocks $m+M\left(v_2\right)$ is based on the speed of block $m$ just prior to the collision with the block $M\left(\mathrm{v}_1\right)$ based on the equation $v_2=(m / m)+M\left(v_1\right)$. This will be discussed further in the chapter on Linear Momentum and Collisions. (b) The loss of kinetic energy as a result of the bonding of the two masses upon impact is stored in the so-called binding energy of the two masses. Calculate the binding energy.

Find the force corresponding to the potential energy U(x) = -a/x + b/x².

Shown below is a small ball of mass $m$ attached to a string of length $a$. A small peg is located a distance $h$ below the point where the string is supported. If the ball is released when the string is horizontal, show that $h$ must be greater than $3 a / 5$ if the ball is to swing completely around the peg.

A

$$5.00 × 10^5-kg$$

subway train is brought to a stop from a speed of 0.500 m/s in 0.400 m by a large spring bumper at the end of its track. What is the spring constant k of the spring?

The sack of mass $m$ has a velocity of $\mathbf{v}_0$ when it is at a height $h$ above the unstretched spring. Determine the maximum compression of the spring if it has a stiffness $k$.

A car in an amusement park ride rolls without friction around the track shown in Fig. . It starts from rest at point $A$ at a height $h$ above the bottom of the loop. Treat the car as a particle. (a) What is the minimum value of $h$ (in terms of $R$ ) such that the car moves around the loop without falling off at the top (point $B$ )? (b) If $h=3.50 R$ and $R=20.0 \mathrm{~m}$, compute the speed, radial acceleration, and tangential acceleration of the passengers when the car is at point $C$, which is at the end of a horizontal diameter. Show these acceleration components in a diagram, approximately to scale.

In the Hunger Games movie (https://openstaxcollege.org/l/21HungGamesclip), Katniss Everdeen fires a 0.0200-kg arrow from ground level to pierce an apple up on a stage. The spring constant of the bow is 330 N/m and she pulls the arrow back a distance of 0.55 m. The apple on the stage is 5.00 m higher than the launching point of the arrow. At what speed does the arrow (a) leave the bow? (b) strike the apple?

Shown below is a box of mass $m_1$ that sits on a frictionless incline at an angle above the horizontal $\theta=30^{\circ}$. This box is connected by a relatively massless string, over a frictionless pulley, and finally connected to a box at rest over the ledge, labeled $m_2$. If $m_1$ and $m_2$ are a height $h$ above the ground and $m_2>>m_1$ : (a) What is the initial gravitational potential energy of the system? (b) What is the final kinetic energy of the system?

A car crashes into a large tree that does not move. The car goes from 30 m/s to 0 in 1.3 m. (a) What impulse is applied to the driver by the seatbelt, assuming he follows the same motion as the car? (b) What is the average force applied to the driver by the seatbelt?

A truck is traveling east at 80 km/h. At an intersection 32 km ahead, a car is traveling north at 50 km/h. (a) How long after this moment will the vehicles be closest to each other? (b) How far apart will they be at that point?

Assume that the force of a bow on an arrow behaves like the spring force. In aiming the arrow, an archer pulls the bow back 50 cm and holds it in position with a force of 150 N. If the mass of the arrow is 50 g and the "spring" is massless, what is the speed of the arrow immediately after it leaves the bow?

Tarzan grabs a vine hanging vertically from a tall tree when he is running at $9.0 \mathrm{~m} / \mathrm{s}$. (a) How high can he swing upward? (b) Does the length of the vine affect this height?

The potential energy function for either one of the two atoms in a diatomic molecule is often approximated by $U(x)=-a / x^{12}-b / x^6$ where $x$ is the distance between the atoms. (a) At what distance of seperation does the potential energy have a local minimum (not at $x=\infty$ )? (b) What is the force on an atom at this separation? (c) How does the force vary with the separation distance?

A cat's crinkle ball toy of mass 15 g is thrown straight up with an initial speed of 3 m/s. Assume in this problem that air drag is negligible. (a) What is the kinetic energy of the ball as it leaves the hand? (b) How much work is done by the gravitational force during the ball's rise to its peak? (c) What is the change in the gravitational potential energy of the ball during the rise to its peak? (d) If the gravitational potential energy is taken to be zero at the point where it leaves your hand, what is the gravitational potential energy when it reaches the maximum height? (e) What if the gravitational potential energy is taken to be zero at the maximum height the ball reaches, what would the gravitational potential energy be when it leaves the hand? (f) What is the maximum height the ball reaches?