While curling, you push a rock for 1.70 m and release it when it has a speed of 1.70 m/s. It continues to slide at constant speed for 0.700 s and then hits a rough patch of ice. It finally comes to rest 9.20 m from where it was released.

To solve this problem, we can break it down into two parts: the motion of the rock before it hits the rough patch of ice and its motion after hitting the rough patch of ice.

Before hitting the rough patch of ice:
1. Use the equation of motion to determine the time it takes for the rock to reach the constant speed of 1.70 m/s.
- We know that the initial speed of the rock is zero (since it is being pushed and released) and the final speed is 1.70 m/s. Therefore, we can use the equation: v = u + at, where v is the final speed, u is the initial speed, a is the acceleration, and t is the time. In this case, we have v = 1.70 m/s and u = 0 m/s, so the equation becomes 1.70 = 0 + a * t. Rearranging the equation, we get t = 1.70 / a, where a is the acceleration. Since the rock slides at a constant speed, the acceleration is zero. Therefore, t = 1.70 / 0 = undefined.

This means that the rock reaches the constant speed of 1.70 m/s instantaneously. So, it takes no time to reach that speed.

2. Calculate the distance the rock traveled before hitting the rough patch of ice.
- The distance traveled can be calculated using the equation: s = ut + (1/2)at^2, where s is the distance, u is the initial speed, t is the time, and a is the acceleration. Since the acceleration is zero, the equation simplifies to s = ut. In this case, the initial speed is 0 m/s, and the time to reach the constant speed is 0 seconds. Therefore, the distance traveled before hitting the rough patch of ice is also 0 meters.

After hitting the rough patch of ice:
3. Use the equation of motion to determine the time it takes for the rock to come to rest.
- The equation we need to use is v^2 = u^2 + 2as, where v is the final speed, u is the initial speed, a is the acceleration, and s is the distance. In this case, the final speed is 0 m/s (since the rock comes to rest), the initial speed is 1.70 m/s, and the distance is 9.20 m. Rearranging the equation, we get a = (v^2 - u^2) / (2s). Substituting the values, we have a = (0 - (1.70)^2) / (2 * 9.20).

4. Calculate the time it takes for the rock to come to rest.
- The equation for the final speed is given as v = u + at. Since the final speed is 0 m/s and the initial speed is 1.70 m/s, we have 0 = 1.70 + a * t. Rearranging the equation, we get t = -1.70 / a.

5. Substitute the value of 'a' into the time equation.
- Plugging the value of 'a' from step 3 into the time equation from step 4, we can find the time it takes for the rock to come to rest: t = -1.70 / ((0 - (1.70)^2) / (2 * 9.20)).

6. Calculate the time the rock slides at constant speed.
- The total time the rock slides is the sum of the time it takes to reach the constant speed (which is 0 seconds) and the time it takes to come to rest. Therefore, the total time is 0 + t.

Using these steps, you can find the time it takes for the rock to come to rest and the total time it slides at constant speed.