# Physics Exercises/Kinematics in One Dimension

Kinematics in one dimension involves motion where the position can be represented by a single number. Motion in other directions (if any) are ignored. For solving kinematics problems without calculus, use of following equations is necessary (these can be derived using calculus):

{\begin{aligned}v_{f}&=at+v_{i}\\x_{f}&=(1/2)at^{2}+v_{i}t+x_{i}\\v_{f}^{2}&=v_{i}^{2}+2as\end{aligned}} .

## Without Calculus

1. A man fires a rock out of a slingshot directly upward. The rock has an initial velocity of 15 m/s. How long will it take for the rock to return to the level he fired it at?
2. It takes a man 10 seconds to ride down an escalator. It takes the same man 15 s to walk back up the escalator against its motion. How long will it take the man to walk down the escalator at the same rate he was walking before?
3. A man riding a bicycle at a speed of 25 km/h passes a parked car. When the bicycle is 100 m ahead of the car, the car starts. The car instantly accelerates to a speed of 50 km/h, and subsequently maintains that speed. Find out when and where the car passes the bicycle, using both a graphical method and an arithmetical method.
4. Cinderella leaves the ball at quarter to twelve in a coach travelling at a speed of 12 mph. Five minutes later the prince also leaves the ball following her. Calculate the speed at which the prince reaches her just as the clock strikes twelve.
5. Two sporty snails are having a race. Because one of them is a famous sprinter, the other is given a head start of 1.0 meters. 15 minutes after the start of the race the sprinter catches up with the slower one. The sprinter was creeping at a speed of 60 cm/min. Calculate the speed of the slower snail.
6. Miss Anna Litical wants to meet a friend living 150 miles away. She starts at 8:30 with an average speed of 50 mph. Her friend starts 30 minutes later travelling at a speed of 35 mph. When and where do they meet?

## With Calculus

1. Starting with the definitions of velocity and acceleration, derive the kinematics equation for constant acceleration x = x0 + v0t + (1/2) a t2.
2. A cockroach starts 1 cm away from a wall. It starts running in a line directly away from the wall with a velocity of 1 cm/s, acceleration of 1 cm/s^2, jerk (d3x/dt3) of 1 cm/s3, d4x/dt4 of 1 cm/s^4, and so on. How far is the cockroach from the wall after 1 s?