give me example exam type questions of how to find the time an object travels and the distance it travels along a surface mechanics type question

Here are example exam-type questions on finding the time and distance an object travels along a surface in mechanics:

### Example 1: Constant Velocity
**Question:**  
A car is moving at a constant speed of 20 m/s along a straight road. How far does the car travel in 15 seconds? How long does it take for the car to travel 500 meters?

**Solution Approach:**  
1. Use the formula \( d = v \times t \) to find the distance.
2. Rearrange the formula to \( t = \frac{d}{v} \) to find the time.

### Example 2: Constant Acceleration
**Question:**  
A cyclist starts from rest and accelerates at a constant rate of 2 m/s² along a straight path. How far does the cyclist travel in 10 seconds? How long does it take to travel 50 meters?

**Solution Approach:**  
1. Use the kinematic equation \( d = v_0 t + \frac{1}{2} a t^2 \) where \( v_0 = 0 \) (initial velocity).
2. To find time, solve \( d = \frac{1}{2} a t^2 \) for \( t \).

### Example 3: Friction and Incline
**Question:**  
A block slides down a frictionless incline of 30° with an initial velocity of 5 m/s. The incline is 10 meters long. How long does it take for the block to reach the bottom? How fast is the block moving when it reaches the bottom?

**Solution Approach:**  
1. Find acceleration using \( a = g \sin \theta \).
2. Use \( v^2 = v_0^2 + 2ad \) to find the final velocity.
3. Use \( v = v_0 + at \) to find the time.

### Example 4: Deceleration on a Rough Surface
**Question:**  
A car moving at 30 m/s applies brakes, causing a constant deceleration of 5 m/s². How far does the car travel before coming to a stop? How long does it take for the car to stop?

**Solution Approach:**  
1. Use \( v^2 = v_0^2 + 2ad \) to find the distance \( d \) where final velocity \( v = 0 \).
2. Use \( v = v_0 + at \) to find the time \( t \).

### Example 5: Circular Motion
**Question:**  
A car is moving along a circular track of radius 50 meters at a constant speed of 10 m/s. How long does it take to complete one full circle? What is the distance traveled during this time?

**Solution Approach:**  
1. Use the formula for the circumference \( C = 2 \pi r \) to find the distance.
2. Time to complete one circle is \( t = \frac{C}{v} \).

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### Follow-Up Questions:
1. How would the results change if friction is introduced in Example 3?
2. What is the difference between solving for time and distance in uniform vs. non-uniform motion?
3. How does the angle of incline affect the acceleration and time in Example 3?
4. How do you find the maximum distance an object can travel before stopping when decelerating?
5. How do you approach problems involving varying acceleration, such as Example 4 with additional forces?

### Tip:
Always carefully identify the type of motion (constant velocity, constant acceleration, or varying forces) as this will dictate the equations and methods used to solve the problem.

Explore example exam-type questions in mechanics, focusing on calculating time and distance an object travels along a surface. Includes scenarios of constant velocity, acceleration, circular motion, and more. Perfect for high school and college students.

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