Kinematics in one dimension

A car covers a straight stretch of $30\mathrm{km}$ on the way out at the constant speed of $60km/h$ and returns to the starting point at the constant speed of $40km/h$. Compute the average speed over the whole trip and the average velocity.

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Two trains travel on straight parallel tracks, toward each other. At the initial instant train $A$ is at station $P$ and travels at $30m/s$, train $B$ is $3.0\mathrm{km}$ from $P$ and travels at $20m/s$. Determine after how much time and at what distance from $P$ the two trains pass each other.

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A cyclist passes a point $O$ at the constant speed of $8.0m/s$. A second cyclist starts from $O$ $5.0\mathrm{s}$ later, in the same direction, at the constant speed of $10m/s$. Determine the instant, counted from the passage of the first cyclist, and the distance from $O$ at which the second catches up with the first.

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A moped starts from rest and reaches the speed of $30m/s$ in $12\mathrm{s}$ with constant acceleration. Compute the acceleration and the distance traveled.

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A train travels at $108km/h$ and brakes with constant deceleration, stopping in $450\mathrm{m}$. Compute the magnitude of the deceleration and the duration of the braking.

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A car travels at $72km/h$. The driver sees an obstacle and reacts after $1.0\mathrm{s}$, during which the velocity remains constant; then the driver brakes with a constant deceleration of magnitude $8.0m/{\mathrm{s}}^2$. Compute the stopping distance.

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An object is dropped from rest from the top of a tower $80\mathrm{m}$ high. Neglecting air resistance, compute the fall time and the speed with which the object hits the ground.

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A ball is thrown vertically upward from the ground with velocity $12m/s$. Neglecting air resistance, compute the maximum height reached and the time after which the ball returns to the ground.

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From the roof of a building $20\mathrm{m}$ high a stone is thrown vertically downward with velocity $5.0m/s$. Neglecting air resistance, compute the velocity with which the stone hits the ground and the time it takes.

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A cart starts from rest and its velocity increases uniformly up to $10m/s$ in $5.0\mathrm{s}$; it remains constant for $10\mathrm{s}$ and then decreases uniformly to zero in $5.0\mathrm{s}$. Compute the distance traveled and the average velocity.

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A point moves along the $x$-axis with equation of motion $x(t)=t^3-3t^2-9t+4$, with $x$ in meters and $t$ in seconds, for $t\ge 0$. Determine the instant at which the point reverses its motion, and the position and the acceleration at that instant.

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A point moves along the $x$-axis with acceleration $a(t)=6t$, with $a$ in $m/{\mathrm{s}}^2$ and $t$ in seconds. At the instant $t=0$ it is at $x_0=1\mathrm{m}$ with velocity $v_0=-3m/s$. Determine the equation of motion and compute the position and the velocity at the instant $t=2\mathrm{s}$.

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A body moves in a straight line with initial velocity $20m/s$ and is slowed down with acceleration $a=-kv$, with $k=0.50{\mathrm{s}}^{-1}$. Compute the time needed for the velocity to drop to $5.0m/s$ and the distance traveled in that time.

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