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“There’s No Time To Explain” is an expression typically used to alert someone of an unplanned event or destination that requires immediate attention. Outside of its colloquial usage, it is often featured in image macros depicting oddly posed animals or people in absurd looking vehicles.
In April 2009, the phrase was also featured in an episode of Charlie the Unicorn. One of its earliest known instances of image macro was posted on I Can Has Cheezburger on November 3rd, 2009, showing an orange cat sitting on what appears to be a backpack (shown below).
Prior to its appearance in image macros, the phrase “There’s No Time to Explain” had been used extensively in movies and television shows to explain a sudden jump between locations or storylines, including in several episodes of Doctor Who and Star Trek: The Next Generation, as well as Woody Allen’s 1991 black-and-white crime film Shadows and Fog. A TV Tropes  page about the phrase’s uses in media was created in February 2009.
In January 6th, 2011, an indie game by the title of the same phrase was released on Newgrounds. The game’s opening sequence starts out with the main character’s future self crashing through the main character’s house stating the words: “There’s no time to explain. Follow me to…”
The game garnered such support that tinyBuild Games launched a Kickstarter campaign on April 16th to fund full production of the game, which follows an unnamed protagonist as he chases his future self who was kidnapped for an unknown reason. They successfully raised $26,086 of their $7000 goal on May 26th and the game was released in August.
On August 16th, 2011, a series of image macros featuring the caption was featured on the viral-content site Buzzfeed. Another set of image macros was then posted on Funny or Die a year later on March 7th, 2012. Other instances have have also been posted on the Cheezburger Network, MemeCenter, Tumblr, and FunnyJunk. As of October 2012, there are 120 results for “There’s No Time to Explain” in the /r/funny subreddit.
Search Interest for “no time to explain” spiked during the Summer of 2011, around the same time as the release of the flash game.
The Next Generation Transcript – Interface | LAFORGE: But how can that be? I mean, how is it possible? SILVA: There’s no time to explain. We have to go down.
Funny or Die – 18 Pictures of… Shit, There’s No Time Explain, Get In