describe and explain motion due to a uniform velocity in one direction and a uniform acceleration in a perpendicular direction

Equations of Motion

Uniform Velocity in One Direction

When an object moves with a constant speed in a straight line, its velocity vector stays the same. We can describe its position after time \$t\$ with the simple equation

\$x = x0 + v\,t\$

where \$x0\$ is the starting position and \$v\$ is the constant velocity.

• Example: A skateboarder glides down a flat ramp at a steady speed of 5 m/s.
• After 4 seconds, the skateboarder has moved
  
  \$x = 0 + 5 \times 4 = 20\,\text{m}\$

Uniform Acceleration Perpendicular to the Velocity

Now imagine the skateboarder starts to jump off the ramp. While the horizontal motion remains constant, the vertical motion is now affected by gravity, which gives a constant downward acceleration \$a_y = -9.81\,\text{m/s}^2\$.

1. Horizontal component:
   
   \$x(t) = v_x\,t\$
2. Vertical component:
   
   \$y(t) = y0 + v{y0}\,t + \tfrac{1}{2} a_y\,t^2\$

Because the two components are independent, we can treat them separately and then combine the results.

Projectile Motion Example 🚀

Suppose a ball is thrown horizontally from a height of 2 m with an initial speed of 10 m/s. The initial vertical velocity is 0 m/s.

Time to hit the ground can be found from \$y(t)=0\$:

\$0 = 2 + 0 \cdot t - \tfrac{1}{2} \times 9.81 \times t^2\$

Solving gives \$t \approx 0.64\,\text{s}\$. During this time, the horizontal distance travelled is:

\$x = 10 \times 0.64 \approx 6.4\,\text{m}\$

Key Takeaways 🎯

• Uniform velocity means the position changes linearly with time.
• Uniform acceleration perpendicular to that velocity adds a quadratic term to the other coordinate.
• Because the two directions are independent, we can solve for each separately.
• Projectile motion is a classic example of this combined motion.