Question

velocity down different ramps
20 degrees 30 degrees 40 degrees
0.5 seconds after release 1.5 m/s 1.7 m/s 1.8 m/s
1.0 seconds after release 2.9 m/s 3.3 m/s 3.5 m/s
1.5 seconds after release 4.3 m/s 4.9 m/s 5.1 m/s
the table indicates the velocity of a marble as it is released down 3 ramps. the 3 ramps have different angles. the same marble was used for each test.
what does the table suggest about the acceleration of the marble and why?
a
at steeper angles, the force is greater creating a greater acceleration
b
at steeper angles, the force is greater creating a smaller acceleration
c
at steeper angles, the force is smaller creating a greater acceleration
d
the force of gravity is a constant and therefore the data in the table must be flawed

Explanation:

To determine the answer, we analyze the relationship between ramp angle, force, and acceleration. Acceleration \( a=\frac{\Delta v}{\Delta t} \). For each time interval (0.5, 1.0, 1.5 s), we calculate the change in velocity (\(\Delta v\)) for each angle:

• 0.5 s interval:

20°: \( 1.5 \, \text{m/s} \)
30°: \( 1.7 \, \text{m/s} \) (Δv = 0.2 m/s)
40°: \( 1.8 \, \text{m/s} \) (Δv = 0.1 m/s from 30°? Wait, no—wait, the table is velocity at each time. Wait, actually, for a given time (e.g., 0.5 s after release), the velocity is higher for steeper angles. Let's check acceleration at 0.5–1.0 s:

For 20°: \( \frac{2.9 - 1.5}{1.0 - 0.5} = \frac{1.4}{0.5} = 2.8 \, \text{m/s}^2 \)
For 30°: \( \frac{3.3 - 1.7}{0.5} = \frac{1.6}{0.5} = 3.2 \, \text{m/s}^2 \)
For 40°: \( \frac{3.5 - 1.8}{0.5} = \frac{1.7}{0.5} = 3.4 \, \text{m/s}^2 \)

Steeper angles (40° > 30° > 20°) have higher acceleration. The component of gravity along the ramp (the force causing acceleration) is \( F = mg\sin\theta \), where \( \theta \) is the ramp angle. As \( \theta \) increases, \( \sin\theta \) increases, so the force \( F \) increases. By Newton’s second law (\( F = ma \)), a greater force (with constant mass) leads to greater acceleration. Thus, steeper angles mean greater force and greater acceleration, matching option A.

Answer:

A. At steeper angles, the force is greater creating a greater acceleration