MCQ Unit IV: System of Particles and Linear Momentum

A ball of mass  $m$   is dropped from rest from a height  $h$   and collides elastically with the floor, rebounding to its original height. What is the magnitude of the average force applied by the floor on the ball during the time the ball is in contact with the floor?

• $zero$

• $mg$

• $2mg$

• $4mg$

• It cannot be determined without knowing the length of time that the ball is in contact with the floor.

A block of mass  $m$ is pulled by a rope along a horizontal surface with negligible friction. For time  $t>0$ , the magnitude of the acceleration of the object as a function of time is given by the equation  $a=Ae^{-kt}$ , where  $A$   and  $k$ are positive constants. Which of the following expressions determines the impulse exerted on the block by the rope during the time interval  $0<t<t1$ ?

•  $A\int_0^{t_1}e^{-kt}dt$  

•  $mA\ \frac{d}{dt}\left[e^{-kt}\right]_{t_1}$  

•  $A\ \frac{d}{dt}\left[e^{-kt}\right]_{t_1}$  

•  $mAt_1\left[e^{-kt}\right]$  

•  $mA\int_0^{t_1}e^{-kt}dt$  

Cart  $A$   is traveling east when it collides with cart  $B$  , which is traveling north. Cart  $A$   has a mass of 3.05 kg, and cart  $B$   has a mass of 2.10 kg. The two carts travel together as a single object on a horizontal surface at an angle  $θ$   relative to due east, as shown above.

In one trial, the initial speed of cart A is 2.5 m/s and the initial speed of cart  $B$   is 1.5 m/s. The angle  $θ$   relative to east that the carts travel after the collision is most nearly

•  $22^o$  

•  $36^o$  

•  $45^o$  

•  $54^o$  

•  $62^o$  

Three identical spheres are thrown from the same height above the ground. Sphere X is thrown vertically up, sphere Y is thrown horizontally, and sphere Z is thrown vertically down, as shown in figures 1, 2, and 3 above, respectively. All three spheres are thrown with the same speed. Air resistance is negligible.

Assume the spheres collide elastically with the ground. Which of the following ranks the spheres based on the rebound height after they collide with the ground?

• X>Y>Z

• Y > (X = Z)

• Z > Y > X

• ( X = Z) > Y

• (X = Z) > Y

Particle A and particle B, each of mass M, move along the x-axis exerting a force on each other. The potential energy of the system of two particles associated with the force is given by the equation  $U=\frac{β}{r^2}$ , where r is the distance between the two particles and  $β$ is a positive constant. At time t = 0, particle A is located at x = 2D with an initial speed of  $v_o$  to the left, and particle B is at rest at the origin, as shown in the figure above.

At time  $t=T_1$  , particle A is observed to be traveling with speed  $\frac{2v_o}{3}$  to the left. The speed and direction of motion of particle B is

•  $\frac{2v_o}{3}\ to\ the\ left$  

•  $\frac{v_o}{3}\ to\ the\ left$  

•  $\frac{v_o}{3}\ to\ the\ right$  

•  $\frac{2v_o}{3}\ to\ the\ right$  

•  $\frac{5v_o}{3}\ to\ the\ left$  

Particle A and particle B, each of mass M, move along the x-axis exerting a force on each other. The potential energy of the system of two particles associated with the force is given by the equation U = β / r ² , where r is the distance between the two particles and β is a positive constant. At time t = 0, particle A is located at x = 2D with an initial speed of v0 to the left, and particle B is at rest at the origin, as shown in the figure above.

Which of the following equations could be used to find r_MIN, the minimum distance between the masses as particle A approaches particle B?

• $\frac{\beta}{4D^2}\ +\ Mv_0^2\ =\ \frac{\beta}{r_{MIN}^2}\ +\ M\left(\frac{v_o}{2}\right)^2$

• $\frac{\beta}{4D^2}\ +\ \frac{1}{2}Mv_0^2\ =\ \frac{\beta}{r_{MIN}^2}\ +\ M\left(\frac{v_o}{2}\right)^2$

• $\frac{\beta}{4D^2}\ +\ \frac{1}{2}Mv_0^2\ =\ \frac{\beta}{r_{MIN}^2}\ +\ \frac{1}{2}Mv_o^2$

• $\frac{\beta}{4D^2}\ +\ \frac{1}{2}Mv_0^2\ =\ \frac{\beta}{r_{MIN}^2}\$

• $\frac{\beta}{4D^2}\ +\ Mv_0^2\ =\ \frac{\beta}{r_{MIN}^2}\$

A ball of mass m and velocity v contacts a hard surface at a 45° angle and bounces off the surface also at a 45° angle and at speed v, as shown in situation 1. Also shown is the force vector F representing the force exerted by the surface on the ball. Situation 2 shows the same ball moving with the same velocity and contacts a soft surface. The time of contact is greater with the soft surface than the hard surface. The ball bounces off the soft surface at an angle of 30° and with speed v₂<v. Which of the following vectors could represent the force exerted on the ball in situation 2.

• 
• 
• 
• 
• 

Identical blocks 1 and 2 slide on a horizontal surface toward a stationary barrier. At time t = 0, block 1 collides with the barrier and slides backward. The blocks collide at time t = Δt. Assuming friction between the blocks and the horizontal surface to be negligible, which of the following statements is true about the two blocks?

• If the collision between block 1 and the wall is elastic, and the two blocks have an elastic collision, then the sum of the kinetic energy of blocks 1 and 2 before block 1 makes contact with the wall will be equal to the sum of the kinetic energy of blocks 1 and 2 after the collision between the blocks.

• If the collision between block 1 and the wall is elastic, and the two blocks have an inelastic collision, then the sum of the kinetic energy of blocks 1 and 2 before block 1 makes contact with the wall will be equal to the sum of the kinetic energy of blocks 1 and 2 after the collision between the blocks.

• If the collision between block 1 and the wall is not elastic, and the two blocks have an inelastic collision, then the sum of the kinetic energy of blocks 1 and 2 before block 1 makes contact with the wall will be less than the sum of the kinetic energy of blocks 1 and 2 after the collision between the block

• If the collision between block 1 and the wall is not elastic, and the two blocks have an inelastic collision, then the momentum of block 1 before it makes contact with the wall will be equal to the sum of the momenta of the two blocks after the collision between the blocks.

• If the collision between block 1 and the wall is not elastic, and the two blocks have an inelastic collision, then the momentum of block 1 before it makes contact with the wall will be equal to the sum of the momenta of the two blocks after the collision between the blocks.

Three objects of mass m, 2m, and 3m are at positions (0,0), (2,3), and (4,0), respectively, as shown above. The center of mass is located at position

• (8/3,1)

• (1,8/3)

• (16,6)

• (8,2)

• (1,1/2)

A person of mass 3M is standing on the left edge of a long, uniform board of mass M and length L that is floating in water, as shown in the figure above. The person walks slowly to the right edge of the board. The water exerts no drag forces on the board. The position of the center of mass of the board when the person is at the right edge of the board is

• 5L/4

• 3L/4

• L/4

• 0

• −L/4