A cannon fires ball horizontally off the top of cliff at 100m/s.Neglecting friction, how long will it take for the magnitude of the vertical component of its vertical component of its velocity to equal the horizontal component?
the vertical component is 9.8t
So, find t such that
9.8t = 100
To find how long it will take for the magnitude of the vertical component of the velocity to equal the horizontal component, we need to analyze the projectile motion of the ball.
Projectile motion is a combination of horizontal and vertical motion. In this case, the horizontal component of the velocity remains constant at 100 m/s because there is no horizontal acceleration.
The vertical component of the velocity, on the other hand, is affected by gravity. The ball starts with an initial vertical velocity of 0 since it is fired horizontally. It will accelerate downwards due to the force of gravity.
The key to solving this problem is to realize that the magnitude of the vertical component of the velocity will reach the same value as the horizontal component when the ball reaches its maximum height.
At the highest point of its trajectory, the vertical component of the velocity will be zero and then start decreasing as the ball falls back down.
To find the time it takes to reach this point, we can use the equation for vertical displacement in projectile motion:
y = v0y * t - (1/2) * g * t^2
Where:
y is the vertical displacement,
v0y is the initial vertical velocity (0 m/s),
t is the time, and
g is the acceleration due to gravity (-9.8 m/s^2).
Since the ball reaches its highest point when the vertical displacement is zero, we can substitute those values into the equation:
0 = 0 * t - (1/2) * (-9.8) * t^2
Simplifying this equation, we get:
-4.9 * t^2 = 0
This equation tells us that the time taken for the ball to reach its highest point and for the magnitude of the vertical component of its velocity to equal the horizontal component is t = 0 seconds.
Therefore, it takes no time for the magnitude of the vertical component of the velocity to equal the horizontal component, as they are already equal at the beginning of the motion.