Position Equation

Question

a ball starts to roll down a ramp with zero initial velocity. After one second, the velocity of the ball is 2 meters per second. How long does the ramp need to be so that the ball can roll 3 seconds before reaching the end?

Answer 1

The acceleration of the ball can be calculated from the initial information and the data after t=1s

a=(Vf-Vi)/t

a=(2m/s-0m/s)/1s

a=2 m/s²

For t = 3 s and constant acceleration (no external forces in the ball)

s=Vit + at²/2

s= 0(3) + 2(3)²/2

s=9 meters

Answer 2

Two methods presented here:

Using Common Sense:

To go from a speed of zero to 2m/s in 1 second means that at the 2 second mark, the speed will be 4m/s, and at 3 seconds, the speed will be 6m/s. Over the 3 seconds, the average speed will be (0+6)/2 = 3m/s. To distance traveled after 3 seconds going an average of 3m/s is 3(3) = 9 m.

Using Formulas:

a = Δv / Δt = (2 - 0) / 1 = 2 m/s²

v_f = v₀ + at = 0 + (2)(3) = 6 m/s

v_f² = v₀² + 2ad ⇒ d = (v_f² - v₀²) / (2a) = (6² - 0²) / (2*2) = 36 / 4 = 9 m