Planar Motion

Solved Problems

Solution.

We take the derivatives of the coordinates $$ and $$ with respect to time $$

Calculate the particle's speed by the formula

This yields:

Substituting the time value $$ we get

Solution.

First, we compute the particle's velocity:

Then the velocity vector is given by

Differentiating once more, we get the acceleration:

Hence

Substitute $$

Then the magnitude of the acceleration vector is equal to

Solution.

$$ The $$coordinate of the ball is defined by the equation

When the ball hits the ground, the $$coordinate is equal to zero. Therefore, we can write

Solve this equation for $$

The second root gives the time when the ball hits the ground:

$$ The time taken to reach the maximum point can be found from the equation $$ Hence,

Calculate the maximum height:

Solution.

First, we write the velocity components for each ball:

Hence, the velocity vectors are given by

When the velocity vectors are perpendicular, their dot product is equal to zero, so that

Note that both the balls are always on the same height as their vertical velocity components are equal. Therefore, the line $$ is horizontal.

The distance between the balls at the time instant $$ is equal to

Substituting the known values, we obtain:

Solution.

$$ Let's solve the first equation for $$ and substitute it into the second equation:

So the particle's trajectory is the right branch of the parabola

$$ To determine the speed of the particle, we differentiate the coordinates $$ with respect to time $$

Therefore, the particle's velocity vector is given by

Since the speed is the absolute value of the velocity, we have

$$ The velocity vector is written in the form

Therefore, the tangent of the angle $$ is given by

Then