Classical Mechanics - Curved Trajectory

The curvature of Mars is such that its surface drops a vertical distance of 2.0 meters for every 3600 meters tangent to the surface. In addition, the gravitational acceleration near its surface is 0.4 times that near the surface of Earth. What is the speed a golf ball would need to orbit Mars near the surface, ignoring the effects of air resistance?

A. 0.9 km/s
B. 1.8 km/s
C. 3.6 km/s
D. 4.5 km/s
E. 5.4 km/s

(GR9677 #22)

Solution:

y = ½ g_marst²

v_x = x/t 

Given:
y = 2
g_mars = 0.4 g 
x = 3600

2 = ½(0.4)(10)(t)²
t = 1 s
v_x = 3600/1 = 3600 m/s = 3.6 km/s

Answer: C

Classical Mechanics - Kinematics

A particle is initially at rest at the top of a curved frictionless track. The x- and y-coordinate of the track are related in dimensionless units by y = x²/4, where the positive y-axis is in the vertical downward direction. As the particle slides down the track, what its tangential acceleration?

A. 0
B. g
C. gx/2
D. gx/√(x²+4)
E. (gx²)/√(x²+16)

(GR8677 #06)

Solution:

A. FALSE.
The particle slides down the curved track → the tangential acceleration is not zero.

B. FALSE.
The particle slides down the curved track, not in the vertical downward direction (the direction of gravity acceleration) → the tangential acceleration is not equal to g.

C. FALSE.
The unit of gx/2 is not the unit of acceleration.

D. TRUE.
The unit of gx/√(x²+4) = g → the unit of acceleration.

E. FALSE.
The unit of (gx²)/√(x²+16) is not the unit of acceleration.

Answer: D

Calculation:

Classical Mechanics - Kinematics

A 5 kilogram stone is dropped on a nail and drives the nail 0.025 meter into a piece of wood. If stone is moving at 10 meters per second when it hits the nail, the average force exerted on the nail by the stone while the nail is going into the wood is most nearly 

A. 10 N
B. 100 N
C. 1000 N
D. 10,000 N
E. 100,000 N

(GR8677 #08)

Solution:

Force: F = ma

The acceleration when the stone hits the nail: 2ay = v² + v₀²

v₀ = 0 → 2ay = v²

a = v²/2y = 10²/2(0.025) = 2 × 10³ m/s²

F = 5 × 2 × 10³ = 10,000 N

Answer: D

Classical Mechanics - Kinematics

A stone is thrown at an angle 45^o above the horizontal x-axis in the +x-direction. If air resistance is ignored, which of the velocity versus time graphs shown above best represent v_x versus t and v_y versus t, respectively?

	vx versus t	vy versus t
A.	I	IV
B.	II	I
C.	II	III
D.	II	V
E.	IV	V

(GR0177 #24)

Solution:

An object such as a cannonball is fired at an angle to the ground:

• The shape of the path or its trajectory is parabolic
• In the x-direction, the object travels at constant speed
• In y-direction, we treat the object as if it were tossed directly upward (positive velocity) and then returned to the ground (negative velocity → the object reverses the direction of its motion).

Answer: C