Answer
(a) - $10.0^{\circ}$: $a=1.71~m/s^2$
- $20.0^{\circ}$: $a=3.37~m/s^2$
- $30.0^{\circ}$: $a=4.88~m/s^2$
(b) - $10.0^{\circ}$: $a=1.70~m/s^2$
- $20.0^{\circ}$: $a=3.36~m/s^2$
- $30.0^{\circ}$: $a=4.91~m/s^2$
Work Step by Step
$x=x_0+v_0t+\frac{1}{2}at^2$
$x-x_0=v_0t+\frac{1}{2}at^2$
$\Delta x=v_0t+\frac{1}{2}at^2$
(a) We are supposing the cart is released from rest.
- $10.0^{\circ}$: $\Delta x=1.00~m$, $v_0=0$, $t=1.08~s$
$\Delta x=v_0t+\frac{1}{2}at^2$
$1.00~m=0t+\frac{1}{2}a(1.08~s)^2=a(0.5832~s^2)$
$a=\frac{1.00~m}{0.5832~s^2}=1.71~m/s^2$
- $20.0^{\circ}$: $\Delta x=1.00~m$, $v_0=0$, $t=0.770~s$
$\Delta x=v_0t+\frac{1}{2}at^2$
$1.00~m=0t+\frac{1}{2}a(0.770~s)^2=a(0.29645~s^2)$
$a=\frac{1.00~m}{0.29645~s^2}=3.37~m/s^2$
- $30.0^{\circ}$: $\Delta x=1.00~m$, $v_0=0$, $t=0.640~s$
$\Delta x=v_0t+\frac{1}{2}at^2$
$1.00~m=0t+\frac{1}{2}a(0.640~s)^2=a(0.2048~s^2)$
$a=\frac{1.00~m}{0.2048~s^2}=4.88~m/s^2$
(b) $a=g~sin~\theta$
- $10.0^{\circ}$:
$a=g~sin~\theta=(9.81~m/s^2)(sin~10.0^{\circ})=1.70~m/s^2$
- $20.0^{\circ}$:
$a=g~sin~\theta=(9.81~m/s^2)(sin~20.0^{\circ})=3.36~m/s^2$
- $30.0^{\circ}$:
$a=g~sin~\theta=(9.81~m/s^2)(sin~30.0^{\circ})=4.91~m/s^2$
