Question

problem 1. a ball begins from rest at the top of a ramp. the ramp is 2.75 m in length. given that it takes 1.8 s for the ball to reach the bottom of the ramp, calculate the balls final velocity.
problem 2. calculate the acceleration of the ball from problem 1 as it rolls down the ramp.

Explanation:

Step1: Identify the known - values

The ball starts from rest, so $v_0 = 0\ m/s$. The distance traveled $\Delta x=2.75\ m$ and the time taken $t = 1.8\ s$. For Problem 1, we can use the kinematic equation $x=v_0t+\frac{1}{2}at^{2}$ to find the acceleration first and then $v = v_0+at$ to find the final velocity. Since $v_0 = 0\ m/s$, the equation $x=\frac{1}{2}at^{2}$.
$a=\frac{2x}{t^{2}}$

Step2: Calculate the acceleration

Substitute $x = 2.75\ m$ and $t = 1.8\ s$ into the acceleration formula.
$a=\frac{2\times2.75}{1.8^{2}}=\frac{5.5}{3.24}\approx1.7\ m/s^{2}$

Step3: Calculate the final velocity

Use the equation $v = v_0+at$. Since $v_0 = 0\ m/s$, $v=at$.
$v=1.7\times1.8 = 3.06\ m/s$

For Problem 2, we have already calculated the acceleration in Step 2.

Answer:

Problem 1: $3.06\ m/s$
Problem 2: $1.7\ m/s^{2}$